There's a LOT to write on the question "what is fascism?" - many political scientists have tackled that and there are a number of schools that answer that question differently.

But as for the question within your question, "Was Nazism left wing?", that comes up quite a bit, and you might want to start by checking out this answer by u/kieslowskifan, and this answer by u/G0dwinsLawyer, plus this thread with some info from u/TonyGaze and u/RicardoHuch.

For background on the differences between Italian Fascism and German National Socialism, check out this answer and this answer by u/Klesk_vs_Xaero.

As other answers have said, fascism is notoriously difficult to place on a simplistic two-dimensional political spectrum. This is largely for two primary reasons: Firstly, because fascism presents itself as a "Third Position" between capitalism and socialism/communism; secondly, because fascist ideologies in practice are not based upon a real theoretical basis but rather upon lies and contradictions dependent on immediate political circumstances. In the case of this latter element, the original pseudo-intellectuals who attempted to create a theoretical basis for a fascist movement were rapidly marginalized and pushed into irrelevance - they were left behind by the far more important "mobilizing passions" (as Paxton called them) that shaped fascist movements more readily.

There are several dueling definitions of what fascism, as a general ideology, can be defined as. The most useful resources, in my personal opinion, are Robert Paxton's book Anatomy of Fascism and Umberto Eco's essay Ur-Fascism.

In his conclusion, Paxton defines fascism in brief as such:

Fascism may be defined as a form of political behavior marked by obsessive preoccupation with community decline, humiliation, or victimhood, and by compensatory cults of unity, energy, and purity, in which a mass-based party of committed nationalist militants, working in uneasy but effective collaboration with traditional elites, abandons democratic liberties and pursues with redemptive violence and without ethical or legal restraints goals of internal cleansing and external expansion.

In general, Paxton claimed that fascism could only arise in conditions of national humiliation and crisis, distinctly as a violent response to an organized left. In the two definitive examples that can be defined as real fascism - Fascist Italy and Nazi Germany - the fascists gained initial momentum as thugs to be exploited as foot soldiers in street violence against communists: the squadrismo in the Po Valley used as strikebreakers for Northern Italian industrialists, and the Nazi brownshirts against the KPD. The Italian blackshirts marched on Rome in 1922 - King Victor Emmanuel III could have turned the army on Mussolini and his allies, but instead chose to give him power; this in the face of a powerful communist movement in post-war Italy (the Biennio Rosso had occured in 1919-1920).

The Nazis, after a period of crippling weakness after the failed Beer Hall Putsch, returned to mass popularity in the wake of the Great Depression. The Depression destroyed the legitimacy of the liberal parties and the Social Democrats, rising the popularity of the extremist anti-republic parties - the Nazis and the Communists. With fear of an imminent (but phantom) revolution on the rise and the appointed conservative chancellors unable to reassert proper government authority, the traditional elites ultimately threw in with the Nazis in order to suppress the left and restore stability. More on this later.

Paxton has another vitally important contribution to fascist ideology - that fascist movements are defined as "movements" precisely because the term "movement" gives the ideology a certain vitality, a certain "constant motion" that gives rejuvenation and new life to its followers. By very nature, fascism is self-destructive - it must constantly confront and overcome crises, or it will either be destroyed (as in the case of National Socialism and Italian Fascism) or it will degenerate into a more generic authoritarianism (as in the case of Spanish Falange or Pinochet in Chile). Fascist movements must unceasingly strive toward an unattainable, even utopian, end goal - the contradictory and impossible nature of the goal is part of what defines a fascist movement, because to attain it would mean the end of the fascist movement's reason for being. In power, fascism is an unrestrained ultranationalist Id, a passionate and hyper-aggressive mobilization of a specific group toward an impossible ideal.

These were what Paxton identified as the "mobilizing passions" of fascist movements:

• A sense of overwhelming crisis beyond the reach of any traditional solutions;

• the primacy of the group, toward which one has duties superior to every right, whether individual or universal, and the subordination of the individual to it;

• the belief that one's group is a victim, a sentiment that justifies any action, without legal or moral limits, against its enemies, both internal and external;

• A dread of the group's decline under the corrosive effects of individualistic liberalism, class conflict; and alien influences;

• the need for authority by natural chief always male), culminating in a national chieftan who alone is capable of incarnating the group's historical destiny;

• the superiority of the leader's instincts over abstract and universal reason;

• the beauty of violence and the efficacy of will; when they are devoted to the group's success;

• the right of the chosen people to dominate others without restraint from any kind of human or divine law, right being decided by the sole criterion of the group's prowess within a Darwinian struggle

While Paxton's analysis of fascist movements is more comprehensive, Eco's Ur-Fascism is similarly important because it identifies very similar elements of fascist ideologies. Eco makes the important distinction that fascism is difficult to define because fascist ideology varies depending on its location of origin - Nazism in America, for example, will not go far because it is a foreign import of what is supposed to be a nativist, ultra-nationalist ideology; an American fascism will be distinct and different from Nazism. But Eco singles in upon fourteen elements that are in common among fascist ideologies, what he calls the titular "ur-fascism". To paraphrase:

1. Cult of syncretic tradition: The wedding of mythologized "traditional values" with certain modern principles (ie industry and military technology)

2. Rejection of modernism: A rejection of modernist political, economic, and artistic theories in favor of more syncretic or Third Position theories such as "Blood and Soil"

3. Cult of action for action's sake: Thinking is effeminate. Thinking is treason. Action is beautiful. Violence is beautiful. Fascist movements must act, unthinkingly and unceasingly, in order to sustain themselves

4. Disagreement is treason: Critical thinking must be stamped out. The intellectuals who do not conform must be persecuted and driven out.

5. Fear of difference: Scapegoat the enemies internal and external. Some Other must be to blame for the problems of the race and nation.

6. Ultra-Nationalism and Obsession With A Plot: There is no such thing as class differences, only the nation and race. All others are the Other, the Enemy, and they are secretly plotting against us both within and without.

7. An enemy who is both too strong and too weak: The Other plotting against us is a sinister conspiracy that threatens our race and nation existentially, but is so weak that if we as a race and nation are united against them can be defeated easily.

8. National/racial humiliation: The enemy must be so strong and rich as to humiliate the targets of their machinations (but are of course defeatable if we band together)

9. Life is permanent warfare: Social Darwinism in the extreme. Pacifism is trafficking with the enemy. But of course, this contradicts with the alleged "Golden Age" of peace that will come when the Others are defeated.

10. Contempt for the weak: The race and nation must be a regimented hierarchy. Those who are below you are below you because they are weaker than you. Thus, they are worthy of contempt. Those higher in the hierarchy despise their underlings, and believe they deserve death.

11. Cult of death: Everyone is educated to become a hero. The most heroic feat is the glorious death in service of the race and nation. Thus, everyone is encouraged to pursue a glorious death as the highest ambition. Fascism is self-destructive.

12. Machismo: Rampant sexism and rigid gender roles. Women are only good for breeding the master race and maintaining the home.

13. Selective populism: A "will of the people" is a fiction, a spectacle maintained to assure the Party popular legitimacy. There is no need for democracy - the leader represents the "will of the people", he is their interpreter, and is infallible.

14. Newspeak: If disagreement is treason, we must create a new language to stamp out disagreement. The Nazis used many euphemisms to cover up the Holocaust in official documentation.

Fascist ideology has another core element to it. Fascist ideology views the state as a very literal body politic. The enemies of the race and nation - racial/ethnic scapegoats or leftists of some flavor - are viewed as a literal infection, a sickness in the body that must be purged to restore its health. While fascist ideologies are generally collectivist, they are very distinct from leftist ideologies in a very significant way - where leftist ideologies view politics in terms of class conflict, fascist ideologies replace the concept of class with race or ethnicity. This radically changes everything. Socialists/communists view society as a conflict between classes, with socio-economic forces pitting one class against another for social supremacy; fascists view society as an interconnected body, with capitalists and workers needing to work in harmony to promote the healthiest body politic. They hold down class conflict via violent coercion, primarily of the working class. Thus, fascism can also be seen as the ultimate form of class collaborationism - it is collectivism in that it is the forced collaboration of the capitalist and working classes for the "ultimate good" of the race and the nation.

It is not leftist in the slightest.

Political formations, in my mind, share the same nature of constellations. As objects created by our perspective, they can be (and often are) influential, decisive even, and thoroughly useful descriptors – but they are not eternal, nor absolute, nor should we rely on them always and forever to guide our path, especially when stranded on foreign lands. And, as the old saying goes, history is just that.

For this reason I don't like to answer the question of “what fascism was” - nor that of “what fascism is”. But I do regard those as two different questions.

Therefore I also – and R. Paxton explains it better in his Anatomy of Fascism, which sadly I don't have on hand to quote the passage properly – agree that we should not investigate Fascism without engaging with its historical context, because the historical context of a political formation is more than a convenient background for our understanding: it's the actual world inhabited by the men whose actions shaped that political formation.

By which I am not saying that it is not possible to regard “fascism” as a generalized, abstract, ideal category; but that such approach often works to our detriment. Unless at least we are very, very clear about what we are doing. Which is not generally the state of things when one asks a question on the internet – and the two questions: “what was fascism?” and “what is fascism?” come to be mixed together.

I am generally reluctant to apply “fascism” and “anti-fascism” as categories to recent or contemporary events. I find this approach to be problematic, and somewhat a-historical. I know there are people who may disagree on this point; and it's true that historical research and political action are not, and should not be a matter of reciprocal indifference. But, again, that requires a certain degree of intellectual clarity, and furthermore a sound theoretical background that I feel I don't possess. R. De Felice was well known (and notorious to his detractors) for his choice to approach his research of Fascism in the 1960s and 1970s with the starting point of looking at Fascism as any other political formation – rather than something assumed to be “wrong”. A reaction – a revision – to both the “fascism-reaction” and the “fascism-disease” narratives. To abandon certain “ideological” vestments (a very abused word) and see Fascism for what it was – and one must realize that this issue was very much a contentious one at a time when many intellectuals and academics had been part of the anti-fascist or fascist experience. Personally I do not regard this approach as an a-political one. But there were those who did. The feeling that Fascism had to be analyzed in context was real though. Explained Costanzo Casucci (who would later be very critical of De Felice's conclusions, especially in the matter of the founding value of the Italian resistance experience), in the introduction to his anthology on the interpretations of fascism [Fascismo e storia in Il Fascismo; antologia di scritti critici 1961 – also cited by De Felice, R. in Il Fascismo; le interpretazioni dei contemporanei e degli storici 1970], that

fascism belongs to us, it's ours, a product of our history, like it or not, and therefore has to be taken in; but exactly because it's ours, of our country, it does not transcend it, but is transcended by it. For a sort of metaphysical denial we anti-fascists would almost like for fascists to be fascists and nothing else […] so that the Italians, once become fascists, would have ceased to be Italians. […] And yet that's wrong. It was Italy, it was the Italians that, at a certain point, became fascists without ever ceasing to be Italians, to become then democrats or to return to be democrats: the purpose of historiography is the analysis of such a process in its full entirety, without hiatus, without parenthesis, without picking apart its components, but identifying them and connecting them back to the unity of history.

It's more or less under a similar perspective that I have begun investigating the origin and formation of Fascism, and – taking from content I have already posted in the Sunday Showcase threads - to follow week by week the formation of the core elements, the appearance, the rise and political affirmation of fascism in Italy, its evolution and transformation into a more and more authoritarian government, and finally its (perhaps incomplete) incarnation in a totalitarian regime, its eventual crisis, its fall and its last short lived resurrection. And in doing so to give back to the reader neither the incisive truth of certain suggestive summaries of fascism, nor the detailed arguments and well thought out structure of academic literature; but one thing that, for their form, purpose and natural limitations, those outlets can not provide: some insight into the rhythm, the pace, the organic patterns through which a country evolves into a new political and social formation, until it becomes almost impossible to tell apart Italy and Fascism, country and regime, nation and empire.

There are many different reasons why the general public around the world has retained or re-discovered an interest towards fascism; but I must confess that for me – and it's one reason I have almost picked up along the way – those echoes of fascism that many seem to hear in the world outside are perhaps less fascinating that some old, slightly outdated, problem of identity. That is, the relation between fascism and national identity in its various naturalistic, spiritualistic, materialistic and – just for the sake of historical completeness – racial formulations.

Now, the existence of a relation, more or less well defined, between fascism and the Italian social, cultural and political formations is almost a given. What is much less clear though, is the nature of such relation: whether it be causal (the Italian system caused fascism), collateral (fascism was an accident on the way), epiphanic (fascism revealed the true characters of the Italian system), etc.

If the need to interpret, understand and explain fascism was present already since the first months of the progressive affirmation of the new political force in the 1920s; the need to understand it in the form of an Italian phenomenon, and especially as an Italian phenomenon, was less ubiquitous and less unchallenged. In part, it betrays the understandable desire within the historian community to look at fascism as a broader European phenomenon encouraged an approach that abstracted from its characters deriving from the Italian social and political system – a certainly legitimate approach, which nonetheless is always at risk of going back and applying its super-national template of fascism to national fascisms in a prescriptive rather than descriptive manner – and in part the convenience of separating Italy and Fascism, the people and the dictatorship, to treat Fascism as a parenthesis within Italian history.

Going back to Casucci, the issue though, runs a bit deeper than the need for the Italian society as a whole to accept the fascist experience as a part of its national “history-heritage”, and the realization that for the Italian historiographical world Fascism was not only an object of study but also a major component of their formative experience (“the crisis of those generations who had matured during fascism”, as Casucci described it) – at times even of their academical formation. Because, as mentioned before, accepting that a relation exists is not establishing the nature of the relation. I'll keep stealing from De Felice's introductions here but, if – in the words of Angelo Tasca - “to define fascism is to write its history”, then to define such relation is to write a history of the Italian national identity, to see if Fascism was, and to what extent, an expression, an element, an incarnation of the national experience until then, or even a possible formation of an immanent national character.