And both are based on a form of Socialism. The DDR was a satellite state of the Soviet Union and the DPRK is based around Juche, which also is a form of Socialism. Another example would be "The People's Republic of China" which makes it sound like China is a democracy (demos = people) while it is absolutely not.
Also the economic engine of Nazi Germany was centered around a planned economy for the common good (or at least, "the common good" according to the Party); an economic system which can only be found in Socialist countries.
> Nazism was an right-wing authoritarian populist movement much like Trumpism
False, because "Trumpism" doesn't want to end capitalism as Trump is a businessman himself who loves capitalism. Hitler hated capitalism and wanted to get rid of it. Also authoritarian? Since when did Trump dissolve nearly all civil liberties in the US, and made all political parties except the GOP illegal? Because Hitler did exactly that, right after the Reichtstag fire and the Enabling Act. Trump sometimes does some pretty awkward shit, but until now he didn't do ANYTHING closely resembling to what those people and freedom hating assholes in Nazi Germany did.
No, it's not, though (especially early on) the mixed salad of policy proposals from different sources they adopted as a platform did include some originating in socialist movements.
Nazism was an right-wing authoritarian populist movement much like Trumpism, and much like Trumpism some of its early appeal to the working class came from some superficially left-wing prescriptions decontextualized from their ideological roots.
> is entirely left-wing to modern standards
Even less true.
> I have actual textual evidence from the man himself to proof it:
No, you have quotes from Hitler that demonstrate that he tried to use appeal to association with socialism as a sales technique. It's not all uncommon for leaders of authoritarian movements to use the rhetorical association with different ideologies than their own as a promotional technique, as hinted at in the “Democratic” in the name of the former German Democratic Republic and the current Democratic People's Republic of Korea.
> Hitler hated "the Left" (as in Marxist Socialism, but mainly because he didn't believe in "their kind of Socialism") as much as "the Right" (Liberalism/Libertarianism).
Classical liberalism is not the right, though it overlaps the part of the right that Hitler opposed in that some portion of it (basically, the part sometimes distinguished now as neoliberalism plus that labelled right-libertarianism) is part of the 20th-21st Century center-right. The Nazis, Fascists, and similar movements were novel far right groups with some similarities to classical conservatism (but incorporating modernist elements from classical liberalism, and corporatist elements various sources across the spectrum, peppered with populist rhetoric.)