Nazis were Nazis. But not infrequently online arguments will result in accusations along the lines of “Nazis were ${X} so ${X} are Nazis”. Which is incredibly shallow thinking of course. A Nazi could well be a stamp collector, but what made them a Nazi was membership of the NSDAP.
There are ‘interests’ that would be indicative of membership of the NSDAP – ‘racial purity’, anti-semiticsm, etc. but they are specific types of interests and something as innocuous as stamp collecting isn’t indivicative.
But concentrating on Socialism because Socialism appears in the name of the party.
Socialism
The full name of the Nazis party was Nationalsozialistische Deutsche Arbeiterpartei[ (National Socialist German Workers’ Party). The “socialist” appears right there in the name of the party, so of course they were socialists.
Well, no. It isn’t quite that simple.
The first indication is that once the Nazis acquired power, they immediately started suppressing socialists of all varieties – the first imprisoned in Dachau concentration camp were Hitler’s political opponents. Including members of the SDP and KDP (see the list of those killed by the Nazis although not all of those killed were socialists).
There are those who will argue that socialists turn on each other. Certainly the authoritarian ones do have that tendency but not to the extreme that Nazis did. For example, members of the German SPD for forced to join the communist party in East Germany. Even Lenin’s Russia didn’t immediately suppress non-Bolsheviks (Mensheviks and the Socialist Revolutionary Party); they were finally suppressed after each was involved in separate uprisings.
The next indicator are economic policies. This is slightly harder to justify because some of the Nazi government spending during The Depression looks a bit like Keynesian but most of the government spending was very often aligned with Nazi military ambitions. Plus they were always very friendly with corporations and had abolished trade unions.
Finally, the Nazis were originally called Deutsche Arbeiterpartei (German Workers’ Party). After Hitler took over the party, it was renamed to Nationalsozialistische Deutsche Arbeiterpartei in a deliberate attempt to deceive working class voters. Hitler was initially opposed to the name, but he was persuaded to accept it
No serious historian is going to call the Nazis socialists and people who do usually turn out to be far-right idiots trying to demonise the left with quite possibly the silliest argument ever invented.