The only reason national socialism is considered a RIGHT wing thing is because in comparison to the communists in the USSR, the nazis allowed for some degree of free enterprise to exist as opposed to the commies who nationalized everything that was worth something important and respected, to some degree, private property. And hence, on the economical policy, they were a bit more liberal in free enterprise. But the nazis and fascists in general have nationalized a LOT of things in their country. From industry to news papers to imposing draconian laws of all kinds and having secret police to instill terror in the inhabitants to make sure they be quiet and submissive.
That's the only real difference, the economic policy between commies and nazis. Politically it was the same single party rulership, no real elections, no freedom of the press and hence, no free speech. No nothing. They were both autocratic, totalitarian regimes and both indulged in the same brutal crimes. Commies hate anti-ethnic policies. Tons of moldavians for instance have been "asked nicely" to move from their homeland to other parts of the USSR and stay there. Same with numerous other ethnic groups like Ukrainians, poles and many more.
So don't go about saying how the nazis weren't lefties. They were pretty lefty. They were just a bit more to the right of the communists.
The reason the far right stuck is mostly PR reasons. You needed a quick way to refer to something, and not that is the mainstream name for such people. Even though in certain perspectives, they are as far left as far left goes.