Was National Socialism a left-wing or a right-wing movement?

So far as I know, there is no serious debate about this amongst academic historians or social scientists.

However, I learned (again) yesterday, that many American conservatives think the Nazis were "socialists."

I was reminded of this by the incoming Senator Elect of Alabama, Tommy Tuberville, who claimed that Americans liberated Europe from socialism in WWII.

And also by an interchange with several political conservatives insisting that Nazism was a form of socialism.

They advanced four arguments, ranging from completely risible to highly debatable:

1) Nazi = "National Socialist."
2) Hitler and Stalin divided up Central/Eastern Europe.
3) Hitler's policies were identical to Stalin's.
4) Nazism was statism/anti-individualism, ergo leftist.

Setting those points aside for the moment, if Nazism was a left-wing movement, then we would expect some or all of the following propositions to be true:

1. Hitler's rise to power was aided by left-wing parties and his regime was resisted by right-wing parties, rather than the other way around.

2. Conservative leaders and intellectuals were subsequently sent to concentration camps.

3. When Nazi jurists visited the US in order to learn about its race laws, they met with left-wing legal scholars suchas Felix Frankfurter and Louis Brandeis.

4. A key part of Hitler's foreign policy was support for Bolshevism and opposition to Franco and Mussolini.

5. Nazi cultural policies celebrated work and production and denigrated warfare and imperialism.

6. Nazi social policy celebrated women's equality and criticized the cult of German motherhood.

7. During the war, the regime expropriated the means of production from the leading industrialists of the era, such as the Krupp and Siemens families.

It also strongly supported labor unions and opposed forced labor.

8. Nazi ideas were based on the works of leftist intellectuals such as Marx and Proudhon, and leftist professors were promoted to positions of authority within German universities. Works by authors such as Nietzsche and Heidegger were banned and even publicly burned.

9. Nazi officials collected artworks by social realists from the US and Russia.

I could go on, but I'm getting tired of typing.