The fundamental divide between Left and Right is their views of human nature. Leftists believe that human nature is a blank slate, that if you engineer the perfect society, you will also engineer away human nature. Maoism is probably the most extreme example of this, they even tried to socially engineer away differences between the sexes by forcing men and women to act, dress, and work identically.
The Right believes that certain aspects of human nature are innate and immutable, and that no amount of social engineering will change the fact that people, as individuals and as groups, are different and unequal. Nazism took this to the extreme of saying some ethnic and racial differences trumped all else. It's certainly a perversion of rightwing ideology, but nevertheless grounded in rightist thinking.
For Marxists, your social circumstances determined your identity. For fascists, your identity determined your social circumstances. This is why these ideologies were mortal, irreconcilable enemies. The fact that they both used a strong centralized government to implement their ideologies is peripheral - it's a means to an end, not the end.
The Nazis were socialists to the core. Like the Communists, they advocated a classless society.
True the Nazis allowed private ownership of industries, but that was merely because Hitler needed the industrialists to build up his war machine. ...And speculation is that once the war was won, and the Industrialists were no longer needed for the war machine, that they were to be dealt with in the same manner as the Jews.