First of all what "left" and "right" refer to. Originally they were the sitting order in parliament after the French Revolution where different factions grouped together on different sides of the room. And the modern usage of these terms is modeled roughly after their political stances.
So the "right" had the conservatives, traditionalists, often religious people and those believing in a natural order of the old regime (church supported absolute monarchy). While the "left" had the revolutionaries, arguing for freedom, equality and brotherhood.
So abstracting from that apparently modern sociology uses the left-right schema as denoting how much a party, group, person, country and so on is in favor or against social hierarchies.
Where left wingers are opposed to social hierarchies and want equality, while right wingers are in favor of social hierarchies or at the very least are fine with them and consider them unavoidable.
This is a pretty useful distinction that helps differentiating a whole bunch of ideologies and applications of ideologies. So for example if you think of all people as equal in value and reject the idea of supremacy/inferiority then international ideas probably reflect that more than nationalist ideas, which favor the progress of one's own nation of that of everybody else's and which often come with ideas of chauvinism/xenophobia/racism towards real or perceived "outsiders".
So with regards to the fact that the Nazis were national(chauvin)istic/racist/xenophobic/antisemitic/abelist/sexist/... Seriously they pretty much checked the box of all of these discriminatory -isms. So with regards to whether they considered human beings as equal or whether they believed in a hierarchical system the answer is a clear 2nd. They were not just considered "right-wing" but FAR-RIGHT. Like they introduced military hierarchies into nearly all domains of life and they have a tough time finding movements and ideologies to the right of themselves.
Though even though we already placed the Nazis on the left-right scale. Let's also for completion talk a bit more about the problem with that scale. And no it's not technically that there are too few axis and it's theoretically also easy to place economical points of view on that scale. Like in it's original idea socialism, basically coops where the workers of a company also own that company and decide for themselves (democratically) what they work, how they work, how they distribute what has been produced or how they invest it. Is a pretty leftist idea, it treats people as equals and avoids social hierarchies but instead goes out of it's way to ensure that in the workplace.
While capitalism, where you have a owner/worker relation where the owner can command people to their will, and if they are in dire economic conditions even against their will, and thus have a higher place in the social hierarchy is more right wing.
So while the racist, the nationalist and the capitalist are all in favor of social hierarchies, they don't necessarily need to agree on that hierarchy. So for a capitalist it might be structured by capital, for a racist, race takes the cake and for a nationalist it's the nation. So while it's possible for a racist to make use of capitalism to enforce it's racist ideas, a capitalist could seek his own supremacy over other people without caring positively or negatively about race. So it's a can not a must condition and while the Nazis checked many of these boxes at the same time that only intensifies the rating but isn't necessary.
The real problem occurs though if you try to measure theory and praxis at the same time. Such as looking at "real life socialists". Like as said the theory sounds pretty leftist, but the application... a central state which has the monopoly of power both economically and politically, a hierarchy of party bureaucrats, often "in one country" so nationalist rather than internationalist to the point of fascism. That's actually closer to being right wing than left wing.
On the other end individualism can have variations that are pretty left wing, if everybody does their own thing and doesn't try to subjugate others, that's in no conflict to equality and the lack of social hierarchies, in fact it could be one and the same thing. However if one of them takes a supreme position and uses the individual rights to prevent other people from accessing theirs, then you've got a contradiction, a gradient of power and a social hierarchy which isn't necessarily implied by the ideology but existent in the application.
So you often have the problem of parties being described or even self-described as left/right, but in terms of what they are doing there might be a rather meaningful difference between proposed goals and achieved goals. That being said in terms of the Nazis the distinction is pretty easy. Their goals were right wing, their actions were right wing and they called themselves a right wing party.
The reason why there is a socialist in their name is because they attempted to be a national biggest tent party and so they tried to market to the biggest possible group, not to convince them but to tolerate them. And while today and especially in the U.S. not so popular, in 1920s Germany there had just recently been a revolution a reformist socialist party had the government for the majority at the time and even a communist party scored double digit results, so they used the label solely because it was popular and might confuse some less informed people. But from their actual political goals and their understanding of human relations they had no interest at all in equality not in the workplace and not in society.