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The Nazis were left-wing.

They were National Socialists, the same way that the Communists were International Socialists. There was very little difference between the two. Hitler even used the Bolshevik symbol - the Swastika.

The only theoretical difference was that Hitler wanted to control the means of production: whereas the Communists wanted to own the means of production.

Both Fascism and Communism were totalitarian oligarchies - indistinguishable to the people trapped under their jackboots.

They were natural allies ( for instance: cooperating on the division of Poland) - right up to the moment Hitler was stymied in the West and needed to get his hands on some more 'means of production' - and launched Operation Barbarossa.

Fighting the Left doesn't make you Right-wing. In many cases it just means you're fighting over the spoils or getting rid of a rival.

Was Stalin right-wing because he had Trotsky killed?

Was Hitler right-wing because he had Röhm killed?

Were the Bolsheviks right-wing because they wiped out the Mensheviks?

Hope this is helpful.

Exactly. They hated each other because they both were totalitarian rivals--only one of them, in the end, could control the machinery of the state. Mussolini for Pete's sake was one of the leading lights *in the Socialist party* and only left in disgust because he found them insufficiently Nationalist. As for Hitler, check out this interview he gave to George Viereck in 1923. I'll print the relevant quote in its entirety so it's clear this is not out of context:

I met Hitler not in his headquarters, the Brown House in Munich, but in a private home - the dwelling of a former admiral of the German Navy. We discussed the fate of Germany over the teacups.
"Why," I asked Hitler, "do you call yourself a National Socialist, since your party programme is the very antithesis of that commonly accredited to socialism?"
"Socialism," he retorted, putting down his cup of tea, pugnaciously, "is the science of dealing with the common weal. Communism is not Socialism. Marxism is not Socialism. The Marxians have stolen the term and confused its meaning. I shall take Socialism away from the Socialists.
"Socialism is an ancient Aryan, Germanic institution. Our German ancestors held certain lands in common. They cultivated the idea of the common weal. Marxism has no right to disguise itself as socialism. Socialism, unlike Marxism, does not repudiate private property. Unlike Marxism, it involves no negation of personality, and unlike Marxism, it is patriotic.
"We might have called ourselves the Liberal Party. We chose to call ourselves the National Socialists. We are not internationalists. Our socialism is national. We demand the fulfilment of the just claims of the productive classes by the state on the basis of race solidarity. To us state and race are one."

So it's abundantly clear that while he loathed Communism, it was because that movement in his eyes corrupted the original, true socialism of the German race.

National. Socialists. The name fits their politics precisely.