Normally, people use "socialism" synonymously (intentionally or unknowningly) with _international_ socialism. There are plenty of other kinds, and _national_ socialism is one of them.
Obviously, if you're going to name your faction with a preceding adjective like national/international then:
a) it's important to you
b) you're probably not going to get along with the other faction that is named in direct opposition to yours
Which pretty much explains how these two groups of socialists hate each other and why the Nazi's 25 point plan [0] reads 50% nationalist and 50% socialist.
Oh, I see. Names don't matter if they are inconvenient....
Nazis (a literal shortening of Nationalsozialismus) - no they weren't the left.. 'cos Hitler, and Hitler bad.
Freedom is speech, but not 'hate speech'. (Who defines 'hate speech'?) What is hate speech except for a reason to stop people speaking. These are ideological positions couched as morality. And everyone needs moral busybodies who know it all, to tell them what to do, right?
“Of all tyrannies, a tyranny sincerely exercised for the good of its victims may be the most oppressive. It would be better to live under robber barons than under omnipotent moral busybodies. The robber baron's cruelty may sometimes sleep, his cupidity may at some point be satiated; but those who torment us for our own good will torment us without end for they do so with the approval of their own conscience.”
- CS Lewis