Neitzsche explicitly said (in longer terms) "Stay sheltered, to remain free. Put three centuries between yourself and current affairs. Do not know who is on which side of what war." So the idea he would care about what wing you are on politically is kind of ludicrous.
Left vs Right is about an approach to history, so it shifts over time, and it is not really theoretically sound. Therefore it is not a thing that really applies to any morality that attempts to be trans-historical. This goes double for Nietzsche.
At the same time you see him as 'leftist' because of his 'relativism', Nietzsche put a great deal of weight on decent taste and on a certain sort of honor that can only really exist in a 'master' society, one that does not consider everyone anywhere close to equal. (Fascism and the alt-right latch on to this and ignore the whole context.)
He explicitly disparaged the Christian leaning toward helping and supporting everyone -- the thing that currently makes the Left the Left, at least currently, in the U.S. He explicitly despised helping people whose problems you have not experienced first hand, or at least come to know through close friendships -- a standard criticism of the 'elite' Left and our current welfare state, which the actual long-term poor often vote against.
Also, moral relativism exists in the form it currently has in order to hold 'the herd' together, in the form of multicultural democracy. It is the new Christianity, a new dominant religion. And it still encourages submission -- only submission to something more objective and less precise. So he would probably absolutely despise most of the modern Left (three hundred years from now, of course.)
Perspectivism is not relativism because it does not encourage compromise and acceptance. It encourages a sort of intellectual domination, enfolding others' worlds into your own entirely, while maintaining your own position. You are not supposed to temper the idea you are right, you are just supposed to understand everyone else enough to know why you are right.
His succession of Creators do not accommodate variant traditions in a broadly multicultural way, they advance totally contrary values in a way that controls and dominates them. Christianity does not remain Jewish... Judaism did not abide the cult of Baal... This approach to reality is far closer to modern Evangelicalism than it is to any liberal or even most Socialist views.