Maybe better than try to pin fascism or nazism on the right or the left would be to recognize that this whole "right-left" thing is a bit arbitrary and subjective and inadequate to reality. What was left or right in the 19th or early 20th century doesn't bear much resemblance to what's considered left or right today. And right and left may not tell us what's really important to know. Too many different views are liable to be bundled together as right-win or left-wing, and there are too many opportunities for complacency and self-righteous preening over one's own superiority.
Commies were correct to place nazis as rightists because they were: they were the right wing of socialism. That is to say, they were socialists who concerned themselves chiefly with their own country and the surrounding territory. They weren’t out to conquer the world like Marxists. They also let various traditional institutions survive, for instance the family, private business, unions, etc., though under severe regulation. They weren’t quite as leftwing as the leftiest of leftists.
Why the should matter to us anymore than the difference between Bolsheviks and Mensheviks isn’t clear to me. Except commies have exerted undue influence on the contemporary mind, and though it wasn’t always the case since WWII no one wants to be confused for fascist. So not only was the infighting between rival socialist factions expanded to the general consciousness, and not only did they trick us into thinking it was of vital meaning, but somehow nazism was twisted into representing not only the right wing of socialism but the right wing of everything! They got to be ancient slavery, feudalism, 19th century laissez-faire, and everything else, or everything else supposedly bad, in the wayback. Leftwing leftists, meanwhile, got to be the future, progress, etc. In which they were aided by no one ever being told what, exactly, that future was going to look like.
This Big Lie was made plausible, slightly, by Big German Business having partly funded the nazi party, which allowed them to get away with the pitiful “last gasp of capitalism” argument (if it was that, it was capitalists trying to hold on by becoming less capitalistic, not more, which incidentally involves them becoming socialists; of course to commies anything non-Marxist is evil and retrograde even if it is lurching to the left). Also, they talked about Frederick Barbarossa and restoring the Holy Roman Empire, plastered pictures of milkmaids and sheaves of wheat everywhere, and invoked the old Norse Gods. They were so oldfashioned that they jumped back a couple of millenia.
Which makes it hard to tell whether it was oldfashioned or futuristic. I run into the same dilemma with communists, too, maybe even moreso. Read the communist manifesto and tell me it doesn’t sound like a romantic aristocrat looking down on upstart burghers. Hard to say whether they’re attacking industrial society compared to the better past or the brighter future. Except that when you ponder what comes after the temporary dictatorship of the proletariat all that comes to mind is primitive tribalism. There’s a great book on this called “The Lost Literature of Socialism” by George Watson. He went back and read what they actually wrote, rather than what defenders and attackers have been les to believe or deliberately led others to believe they were on about. I can’t even summarize it here, really, because it’s that far away from how I’ve been accustomed to think.
Right-left thinking requires you to think relativistically. Which we do all the time, but can be tricky when people are as interested as they are in the outcome when it comes to politics. Lazy thinking has leftists as “progressive” and futuristic and fighters have oldfashioned. We don’t know what the future holds, we could both go boldly forth and revert simultaneously, like Iran a few decades ago. Most libs now are socialists and stand somewhere just shy of fascism.
On the other hand do conservatives celebrate everything in the past? We so-called in the contemporary U.S. pay homage in words to “the Western tradition” and Judeo-Christian values, but our specific political ideas largely derive from mo further back than the classical liberalism of the enlightenment. Most of us don’t even make it back to the Reagan era, despite big talk, and only isolated nuts pretend like we can return to pre-New Deal days. We stand shy of fascism, too, only less enthusiastically.