The differences in the parties at the time were more economic than social, and regional issues carried more weight than they do now. It was a different political climate. It’s not always as black and white as some would like you to believe. Fascism, for example, does not fit neatly in the linear Left-Right paradigm of today, no matter what the hyperpartisans will tell you.
Both communism and Fascism sought to violently put down Christianity and replace it by the state. Not just religion but every social group was to be attached to the state. You cannot get any more social than that, The movement was called Socialism after all. You would expect this level of social change being championed.
Have you read the 1919 Fascist Manifesto? Minimum wage, progressive taxation, state take over of all arms companies and most of their past profits nationalised and a commitment to defence only, voting rights to be given to the poor, voting rights to be given to women, state take over of church property and abolishing of diocese system et cetera,
This is the ground zero of what they thought of themselves rather than 70 years of Left controlled education projecting it otherwise. If we want to know what Marxism is we read Marx’s manifesto. If we want to know what Fascism is then we read the authors of the Fascist Manifesto, Mussolini and Gentile, the two creators of Fascism.
When Hitler’s National Socialists went to war with Britain the rhetoric in the SS publications was how Britain was an oppressive capitalist system with the London upper class exploiting the poor workers in contrast to the egalitarian social programs of Hitler’s Germany. It is right there in black and white (although in German). When Hitler’s Socialists betrayed the Soviets in their non aggression pact the rhetoric was how the National Socialists would “show the Russians what true Socialism is. There would be no come back for priests and rabbi’s.”
Hitler identified with big state socialism, the destruction of the Christian churches and the victim politics of claiming to be on the side of the oppressed worker. He did call his movement the National Socialist Workers Party after-all,