I’ve been a few months hearing (always from right wing extreme neoliberal people) that Hitler and Mussolini were socialists, that’s why the term Nazi comes from Nationalsozialistische Deutche Arbeiterpartei. I do now that Mussolini had a socialist past but I would like to know in a more profound way the facts why this affirmations are true or false. Thank you.
This response from u/kieslowskifan addresses why the Nazis were called National Socialists, despite being fascist.
The Nazi party and the Italian fascist party were right-wing parties, not socialist parties. There's plenty on this in the FAQ.
Those claims are usually rooted in a fundamental misunderstanding of both Mussolini's political trajectory and the Italian political situation of the early 1920s. Also, given that information on those periods is not exactly a mistery, one needs to be extremely generous not to see those efforts as largely disingenous attempts to project concepts, and even terminology, over a reality where they certainly don't belong. Most of the time, the reason why those statements are wrong, is the fact that they hold to their own definition of "socialism" and insist on projecting it over a reality which doesn't fit their definition in the slightest. Which is also the reason why I think it better to spend some time actually examining what was going on at the time. Anyways, if you have some specific claims in mind, feel free to ask.
I have almost finished adding bits to an answer concerning Mussolini's transition from socialist leader in the early 1910s to leader of Fascism in the early 1920s. I have written about more or less the same thing here and in many other instances I can't locate right now. And here is something on the rise of Fascism in Italy during the early 1920s.