Gorfias said:
The fact is, fascism is a leftist ideology, not of the right. The right wants to be left alone. The left wants to control you.
Hayek is free to have his opinion that fascists were socialists - however, many philosophers, political scientists and economists disagree strongly. With such a lack of consensus, there is no safe way to argue one way or the other. Except, perhaps, the obvious middle way that it truly belongs to both or neither the left or right.
It's worth pointing out that the only parties that struggled to preserve democracy by opposing Hitler's power grab in the early 30s were the left-wing ones, the SPD and the Communists (although the Communists had been forcibly ejected from parliament by then). The entire right and centre-right of German politics at the time ultimately backed Hitler.
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"Left" and "right" wings are variable terms, and hence lack a lot of accuracy. The terms originally refer back to the French Revolution, where the "right wing" represented the status quo, monarchy and aristocracy, and the "left wing" the progessives and radicals. Later on, right wing became associated with capitalism and nationalism, and left wing with socialism.
Neither "right" nor "left" meaningfully discusses totalitarian government. Vast numbers of brutal, repressive dictators (Franco, Pinochet, Syngman Rhee etc.) have been undeniably right wing, and you couldn't say they were interested in leaving people alone. Equally, the left has had plenty of totalitarian states (USSR, China, Cambodia). Many left wing organisations have defended people's freedoms and democracy, as have many right wing ones. Similarly, in terms of international interference, both right and left have at times and in varying ways left their neighbours alone, been actively isolationist, started wars or aggressively meddled in other nations' affairs. There is nothing to choose.
Then also consider social freedoms. If you look back historically, the parties that have tended to oppose discrimination and and egalitarianism have often been left wing. It was generally left or left-leaning parties that enfranchised women, legalised homosexuality, fought against discrimination of racial minorities, etc. Time after time it has generally been the right wing, or large sections of the right wing, that have fought hard to maintain a socially unjust status quo - arguably as is the case in the US currently.
When you say the right wing wants to be left alone, and the left wing wants to control, you would need to view "control" purely in economic terms - yes, the left wants more of an individual's money for the society as a whole. The left wing views empowerment of the individual in ways other than financial.