Well, they were the National Socialist Workers Party.
They certainly were into "social engineering" weren't they?
Lots of commie countries called themselves Democratic Republics though they were neither.
Socialism, to my mind, is simply “all men are created equal” carried one step too far, to include economic equality as well as social and political equality.
“Socialists” have always varied enormously about how they think that goal should be achieved, and what methods they are willing to use to get there. A lot of Americans equate Marxism and Socialism, but of course Marxism is only one of the many variants, albeit the most famous and influential.
If you accept this definition of “socialism,” then Nazism doesn’t qualify. Its goal was to create a permanent hereditary aristocracy of the German people dominating the rest of the world. And of course it’s Fuehrer Principle pretty much put paid to the idea of any real equality within the German people themselves, though it was not supposed to be hereditary.
The Nazis are very difficult to fit into any of the traditional political categories. Italians Fascists, OTOH, are more clearly a variant of socialism.