Fascism was regarded as a "right-wing" movement at least since its resurgence in 1920, when the fascist squads became instruments of the agrarian anti-socialist reaction in the large agricultural regions of the Po River valley.
The Italian fascist movement had had a distinctive anti-socialist tone since its foundation in March 1919, even if Mussolini made an active effort to distinguish between the Official Socialist Party (the "Italian Bolsheviks", the "traitors of the Motherland") and the proletarian masses. Fascism was - he pointed out - against the Official Socialists, "not against the workers".
This was part of his attempts to position his new (at the time absolutely minor) movement somewhere around his general platform for a concentration of the interventionist forces, leaning towards "democratic interventionism", also known as "left interventionism", since the opposite end of the spectrum was hegemonized by the nationalists; which tied back to the proposal for the formation of a "constituent of interventionism", or of a "democratic constituent" which gained some traction during the Spring of 1919.
Despite these attempts, which led to some difficulty for contemporaries to classify early fascism (the program of 1919 followed various points of De Ambris' syndicalism, but had also inherited certain arguments of the so called "national syndicalism") - for instance, the fascists ran for the national elections in 1919 with a "democratic block" in Milan - many of the fascist themes (veteranism, trencherism, productivism, the adoption of the nationalist platform on the matter of Dalmatia and Fiume, anti-socialist violence, a constant vindication of the national right, as in right of the nation, a traditional if somewhat subdued anti-parliamentary and anti-liberal tone) favored a closer relation with other forces of the so called "national radicalism": futurists, arditi, and even (with more than a few incidents along the way) the Nationalist Association.
Taking into account how small fascism was at the beginning, there was already a certain local contiguity with forces of unmistakable right-wing collocation, which had begun to surface at the time of the political elections of 1919.
The major electoral failure - in which no little part was played by the divisive figure of Mussolini himself (the "democratic" formations had rejected his offers, with the argument that his presence in their lists would have cost them the votes of any socialist discontent) who had become, perhaps even beyond his intentions, of anti-(official) socialism after his involvement with the fire of the Avanti! of April 1919 - led to a substantial revision of the fascist program, and actually to a reform of the entire organization. Not that there was much to reform, as by the end of 1919 the fasci counted less than 1,000 members nationwide.
With the end of 1920 and especially 1921, the fasci regrew organically thanks to the support of the agrarian (and later on, even if never reaching the same degree of identification, of the industrial) organizations, which provided funds, a practical raison d'etre, and a strong incentive to adopt a more "national right", "order" platform - even in the context of Fascism's subversive and illegal initiatives. Typical the famous incident which occurred in Ferrara, when the local fascio had published the 1919 program instead of the new one for the administrative elections, prompting the irate reaction of the local basis and an immediate reply that, the 1919 program was "the old one" and the new one was "entirely different".
This incentivized those moments of collaboration with the nationalist and right-wing conservatives which had already appeared in local elections; and also contributed to drive away from the movement those elements whose ideological collocation was more ambiguous (fascism did maintain to a degree an ideological shade of productivism and "anti-capitalism", or maybe better of "anti-establishment", which was nonetheless entirely consistent with the ideological forms of the far right - think of the contemporary attacks against Nitti's "financial Bolshevism" or the more famous ones against the British and American "plutocracies").
As a result, by 1921, most observers looked not only at fascism as a right wing movemenr, but at the perspective of an institutional fascism as a decisive force of the national right.
Additionally, it's worth noting that our post WW2 ideas of right-wing and left-wing (regional differences notwithstanding) are not easy, or much useful, to overlay on past political formations. While certain distinctions remain, others make very little sense and are impossible to translate in modern terms. In fact the opposition between Fascism and Bolshevism was - at least in 1919-21 Italy - a much more prominent and marked one than the traditional one between liberal right and liberal left (or historical right and historical left). So that I wouldn't say that we "begun" calling fascism "right-wing" or communism "left-wing", but rather that those ideas make up a portion of our understanding of the "left-right" opposition. Analogously, the people in the 1920s and 1930s saw these concepts as elements of their understanding and partitioning of the political sphere.
/u/klesk_vs_xaero covered the fascist side really well. I just want to add a bit regarding the communists. This time, I think things are much more straightforward. Communists grew out of divisions arising within the socialist movement dating back to at least 1905, when more radical members of the Second International wanted to abandon the long term/organization building approach of the hegemonic German Social Democrats, in favor of a more direct challenge to the contemporary social order. The most famous of these was of course Lenin, who believed a vanguard party was essential to move things forward while pulling the masses along, but others within or adjacent to the socialist movement (syndicalists, anarchists, etc) believed that at the very least tactics like the general strike should be adopted to better challenge the state. These were all rejected, under the leadership of the German SPD, in the Congress of 1906. This created a fissure between the more nationalist social democrats like the SPD and a smaller radical faction, heavily represented by the Eastern European members of the international, who believed working with the state was an illusion. When World War I broke out less than a decade later, the socialist movement permanently split between majorities who supported the war for various reasons and minorities who opposed it on the grounds of international proletarian solidarity. When the fighting ended, many of these independent socialists (as the German USPD was known) remained unreconciled to the majority parties and would eventually, usually after a lot of revolutionary violence, form the core of the new communist parties, which thanks to the Bolshevik success in 1917, followed a much more centralized and authoritarian line than they might have done, given the prominence of libertarian forms of socialism before the war.
So, communists are on the left because they grew directly out of splits in the socialist movement, and socialists have been pretty unambiguously on the “left” side of the spectrum since the early days of that conception go politics in the early 19th century. But I also think it’s true that one shouldn’t overplay the left-right thing with historical politics.