Fundamentally, fascism is anti-Left.

Yes. The National Socialist Worker's Party (NAZI) has "socialist" in the title, but that's about as far as it went. It was really about attracting the blue collar peasants, and then dosing them with a lot of nationalism (and scapegoating..."it's not your fault, the Jews are keeping you down!"). The USSR was more socialist, and authoritarian (but not, strictly speaking, "communist"). The Third Reich was hardly anything resembling socialism, unless you count creating jobs to build a lot of hardware. The state did not own very much production, but the authoritarian nature allowed them to dictate what was produced (from Volkswagen Beetles to Panthers). Neither Stalin nor Hitler tolerated dissent, but that was because both were authoritarians.

I think it safe to say that the NAZI party very early on adopted right-wing policies, touted nationalism, and mixed in a bit of quasi-religion (sort of a Nordic/Aryan mythology, with Hitler as a "hero" of that mythology).