So, let's do a reality check: Both Hitler and Mussolini pretended to have socialist aspirations as part of their propaganda efforts during their rise to power, largely as a way of encouraging working-class support. But they were unquestionably right wing politically by the time they obtained power, and in fact were viciously anti-left-wing as well.
Those who repeat the "Nazis were socialists" claim are, in fact, falling for (and repeating) Nazi propaganda from the 1920s.
Mussolini was indeed an active socialist at the beginning of his political career. But he was remarkable for shifting his alliances and adjusting his ideology accordingly as he climbed the ladder of power; and by the time he had completed his climb, he was an outspoken and lethal anti-socialist.
And I'm very much looking forward to Niewert (along with probably many, many others) eventually taking his red pen to this.