Fascism, socialism, national socialism and communism are all one and the same - totalitarian tyranny. On the scale of government power versus personal freedom, which is the true scale by which we are to categorize any political philosophy, they all occupy the left side of the scale.
Anarchy, the Complete and total absence of governing authority, occupies the right side. The scale then trends into degrees of libertarianism as you move to the center where you find the republic, or equal rule for all by the law.
The left would love you to believe fascism is not leftist, but in truth you cannot reconcile it that way using the proper means to judge it.
Your definition of right-wing as anarchic is ahistorical.
As you (should) know, the terms right and left originated during the early days of the French Revolution, when the opponents of the ancien regime sat on the left, and its supporters on the right. Those supporters were the very opposite of anarchists.
Republicans (in the American Revolution sense), anarchists, and what we would later call socialists and fascists were all on the Left when the French Revolution started. Since the Left had originally been in favor of change, and the Right against it, those relative positions tended to stick as the spectrum changed.
The original Right quickly was destroyed or driven into exile, with roughly American-style republicans and then various less-extreme groups of leftists replacing them as the Right as time went by.
But the notion that any and all forms of oppressive government are socialist just won’t fly. The Roman and Chinese Empire under their worst emperors weren’t socialist.
Socialism, as such, is at root an economic doctrine. It believes that public ownership of the means of production will bring the Millenium.
Fascism is, I agree, partly socialist in origin, but it also incorporates large elements of the original crown and church ideology, blood and soil ideology, etc. Oddly, given what American conservatives are so often called, of all political ideologies it has perhaps the least in common with American conservatism and its individual rights.
I detest the term Right as applied to American conservatism. It has nothing at all to do philosophically or historically with European crown/church, blood and soil conservatism. While American liberalism and leftism have a direct line of descent from the French Revolution Left, American conservatism descends from British Whiggism, an entirely different ideology.