A nit, the word socialists is actually in the acronym (ie in the name). Fascism actually is a descendant of socialist ideology and Mussolini was a Socialist before inventing Fascism. Concepts of left and right only have meaning in one political unit at one point in time. The axis we use to judge right and left changes from time to time and is different in different places. Also, some fascist policies are similar to policies in socialist societies. And some policies are very different between the two. So the debate implied in your question, like most debates, comes down to the definitions of words.

This is why it isn't all that helpful to base political ideologies on history farther back than one human lifetime. The writers often meant something different than you think they did.