The fact that there were many different inter-squabbling socialist factions can't hide the fact they did all call themselves well, socialist. Whether someone else wants to sniff & say they weren't true socialists leaves us w/ saying they were ok, the bad socialists.
But they weren't even socialists. They had private property in Nazi Germany. They had private entities controlling the means of production. They had wealthy people. They had titled aristocrats and nobles still around. These are not indicators of a socialist, anti-monarchist, or anti-capitalist regime.
It doesn't matter what they called themselves. At one time, there was an entity known as the Holy Roman Empire, which was neither Holy, nor Roman, nor an Empire.
If you always believe what a government calls itself, then you must believe that North Korea is a People's Democratic Republic. Is that what you believe?
fwiw, (& I'm not sure if it makes any difference) Fascism was Italian. The big mantra of both National Socialism and Fascism was that they were not capitalist, they were not Communist, they were the third way between the two. Bottom line it's still hard to somehow say they were conservative right wingers. Lots of folks, even today, call themselves the "third way" between capitalism & communism --and they're no way around it left of center.
The salient features of fascism are nationalism, racism, and/or ultra-patriotism - intense love of nation above all other things. Whether it's socialist, capitalist, authoritarian, or libertarian makes no real difference, as long as love of nation overrides all other considerations.