Finally, a contemporary Argentina question! I'm no Peronism expert but will give it a shot based on my knowledge of more recent events. Others have talked about “what is Peronism” so I’ll focus on giving some context for it. (sorry for the long read)

You're absolutely right that Peronism (peronismo) is arguably one of the most internally contradictory movements of all time, at least from a US/European perspective well used to the familiar left/right divide and some semblance of internal ideological cohesion. However, that's not to say there's no sense to be made of it, especially when we understand it more as a "populist" movement (more on that term and its limits later) within the very particular world of Argentinian political identity.

Before we get to Peron, however, I'll ask you to indulge a sweepingly broad history of the country up until the arrival of Peron. Argentina, in the mid-nineteenth century, undertook a program of "National Organization" aimed at ending the decades of civil war that had followed independence and starting Argentina's new life as a fully European-style nation. The governments of the National Organization and for decades afterwards represented a sort of consensus among Argentina's elite governing class on the importance of modernization, creating and upholding European-style political and social institutions, and encouraging the immigration of Europeans to Argentina, out of a nineteenth-century Social Darwinist view that making a new country as successful as Europe involves filling it with white people (to this day, Argentina is still a very white country, in part as a result of these policies).

However, in 1930, dissatisfaction with the ruling liberal president, and the ascendancy of a new, fascist-tinged nationalist movement led to Argentina's first military coup, which started a new era in Argentina of military involvement in political affairs. In fact, it instilled in Argentina's already conservative military a sense of political duty and political identity, largely tied to the nationalist movement, which would lead to a coup roughly every 10 years between 1930 and 1976 (1930, 1943, 1955, 1962, 1966, 1976).

The 1943 coup was especially noteworthy when we begin to talk about Peron. While there were various reasons for it, a primary one was Argentina's stance as neutral in World War II. Argentina's military, being so tied up in the nationalist movement, had its fair share of fans of the Axis, not helped by the fact that much of the officer class had long admired the Prussian/German Army, and some had even been trained in that tradition. While the US was pressuring Argentina to join the Allies, the military pushed for neutrality (as tacit support for the Axis), and overthrew the government when the president proposed supporting the Allies. It's safe to say, therefore that the "United Officers Group" that spearheaded the 1943 coup was composed of right-wing military men who were acting out of a declared admiration of, and in support of, German and Italian fascism. Peron was one of these officers, and while he may not have been the mustachio twirling Nazi portrayed in the movie Evita, nevertheless admired Mussolini especially and had strong personal ties to Nazi Germany (modern-day Peronists still reject these claims). So, Peron was ostensibly a hard-right military man at the beginning of his career, and maintained some level of fascist sympathies for the whole of his life.

However, upon taking power in 1943, the insufferably ambitious Colonel Peron was given the shittiest job the new administration could find-- head of the newly founded Ministry of Labor, which was expected to be a dead-end job that would prevent him from being a nuisance to his superiors. However, Peron, ever the able political actor, quickly allied with Argentina's growing labor unions, took a very prominent role within the parties of the urban working class, and very quickly the Peron-union alliance became the most powerful force in Argentinian politics, first moving to pass a series of labor reforms (which Peron made sure either bore his name or were closely associated with him personally), then getting Peron promoted first to Minister of War, then to Vice-President. When a shambolic and poorly organized anti-Peronist effort, aided by the United States, got Peron briefly deposed and arrested in late 1945, popular pressure, culminating in a hundreds-of-thousands strong march on the presidential palace, returned him to power, and he won the 1946 presidential elections.

In power, Peron continued to exhibit tendencies that seem contradictory on a left-right axis but all fit in to the larger trend of assuring his power and satisfying his base. In fact, while the term “populist” gets thrown around a lot these days, Peron fit it better than most, as many of his policies in the economic and social sphere were a way to directly satisfy his working-class base as a way to ensure their continued support of him. And this is the way in which Peron can seem “socialist” to our eyes, as he enacted sweeping labor protections and social reforms ranging from minimum wage and safety regulations to voting rights for women, etc. And in this context, Peron seems to be a pretty traditional “socialist” reformer.

However, at the same time as all this, Peron continued to make moves towards strengthening his own executive power, in ways that make him look a lot more like a traditional strongman. Now, this wasn’t without its precedent, as the phenomenon of caudillismo, or strong one man leadership stemming from the Spanish tradition, in Latin America is pretty well documented, and was especially popular in Argentinian nationalism. So while he was empowering workers and protecting their rights, he was also doing everything in his power to rule as a dictator, going so far as to change the constitution in 1949 to give himself sweeping presidential powers, which enabled him to run for and win an unprecedented second term in 1952 (Argentina’s constitution, written in the 1850s after an extended period of dictatorship, had a strict one term limit as one of many checks on presidential power designed to curtail future leaders like Peron). School texts from the period read like an Orwellian horror show, as children would learn grammar by reciting sentences praising Peron or his wife Eva (Evita). Peron’s second term was even more openly dictatorial, and the so-called “Liberating Revolution” of 1955 (another military coup), was explicitly aimed at shutting down Peronism for good, and forced him into exile.

The social conflict that followed Peron’s exile and his return to Argentina in the early 1970s provides a good lens for analyzing his legacy. Argentina, in the late 60s and 70s, was in total social chaos, as various armed political groups vied for assorted political objectives. There were a lot of groups at play, but most importantly there were Peronist groups fighting on different sides. Marxist-Leninst rebels and right-wing paramilitary death squads, some of which were working under orders from Peron himself, both laid claim to the label of “Peronist”, and truth be told both were sort of justified in doing so, as they represented the different parts of his political package.

So, in conclusion, in Peronism we have a movement that has both hard-left and hard-right tendencies, in reflection of a highly complex man. Personally, I think of Peron more as a conservative military strongman than a leftist crusader, as it seems to me that his overtures towards workers were more because they were his avenue to power rather than a genuine interest in the interests of the lower classes. But millions of Argentinians disagree, and the student and union political spheres, especially, are still dominated by Peronist groups (or rather, groups claiming to be Peronist). The contradictions of Peronism, while they have explanations and reasons for being, are still mind-bending contradictions that have to be worked around (or simply paid lip-service to). Maybe the most important legacy of Peronism is that it provided a figurehead for various social and political movements, and that it has given Argentina its defining question of political identity. Just as in the US you’re usually a Democrat or a Republican, or in other parts of the world you’re more left or more right, in Argentina you’re either pro- or anti-Peronist, whatever that means.

If you're interested in more, I'd recommend David Rock's Argentina, 1516-1983 or Luis Alberto Romero's A History of Argentina in the Twentieth Century for general overviews. If you want a more in-depth and politically-focused read, go for Federico Finchelstein's The Ideological Origins of the Dirty War, which is a fascinating history of fascism in Argentina.

TL;DR: The contradictions in Peronism are a direct result of Peron's actions. It's above all a populist movement, with pro-labor and distinctly authoritarian tendencies.