The claim you quoted said "left-wing", it didn't say "Socialist".

And the parts that you didn't quote mentioned that fascism did sometimes describe as a mixture of both left-wing and right-wing ideas, just like "National Socialism" included the word "National" to appeal to right-wing nationalists, and "Socialism" to appeal to left-wingers.

If you want to make a rebuttal to my actual claim, find a place where Nazism or Fascism describes itself as "left-wing" -- just left-wing, not "a response to both left and right" or "a synthesis of both left and right", or indeed "National Socialist".