In 1932 Mussolini wrote (with the help of Giovanni Gentile) and entry for the Italian Encyclopedia on the definition of fascism:
..Fascism [is] the complete opposite of Marxian Socialism
The entire article by Mussolini makes for interesting reading, I would suggest it. I makes point that support either of our propositions.
So, in practice, Communism and fascism obtain their ends via the State, and conservatives and libertarians obtain their ends via the individual. If we separate left and right along who controls us, Fascism and Nazism wind up on the left.