While not a direct answer to your question, u/kieslowskifan posted a fairly in-depth response a few years ago to the question of whether the National Socialist German Workers' Party were indeed socialists.

The short answer is no, not really. British Historian Richard J. Evans addresses this in great detail in the first book of his Third Reich trilogy - The Coming of the Third Reich (October 2003) - pointing out that Hitler began separating his ideology from other German "socialist" movements as early as 1920. Indeed, the NSDAP went as far as to purge their own "socialist" paramilitary, the Sturmabteilung (SA, or Brownshirts), in the Night of the Long Knives in 1934 in order to ensure their own peculiar definition of socialism remained unchallenged within the party.

Evans argues that, while the NSDAP used quasi-socialist rhetoric from the outset to enhance and broaden their electoral appeal, Hitler's - and by extension his party's - definition of socialism remained fluid and cynical in the extreme, meant only to broaden the appeal of his vaguely defined ethnic-German utopia, devoid of class warfare.

Evans' trilogy isn't written for specialists on the subject, and I found it to be fairly accessible getting into the roots of the Nazi party. Worth a read if you're interested in a deeper dive on the subject.

Edit: I've added a more detailed response below that I think more directly addresses the OP's original question regarding Nazi socialism as a post-war talking point.

Evans, Richard J. The Coming of the Third Reich (London: Penguin Books, 2003)

I had read both u/kieslowskifan's and u/fallskjermjeger's and they are both helpful, but I think it is necessary to make some further comments about this topic.

First of all, the idea that the Nazis or the Italian Fascists are somehow socialists is a recurrent and increasingly popular one (thanks to the internet) in American right-wing circles today. It even caught a little bit in Europe for those that try to emulate the US brand of right wing ideas. I hope that is ok to speak a little bit about current ideology because it is important to put the OP's question into a little bit o context. I hope the mods will understand.

While Hayek also mentions that Nazism has socialist roots in the 40's, the idea is somewhat marginal until way latter when it was taken by right-wing conservatives especially in the USA. A part of their narrative is not a historical debate, but rather an ideological one. Their main "scare" is socialism and they try it to portray it as a 1). economic failure, 2) criminal ideology while their neoliberal economical outlook and social conservatism is the way forward. In this context it is easy to portray Marxism-Leninism as meeting the criteria of 1) and 2), but they are reminded that right wing extremism as as criminal as left wing extremism. As ideological split became more and more entrenched in the US, some on the right started to adopt this view that somehow it must be socialism fault for the other criminal ideology of the past century. Because US conservatives put a lot of emphasis on economics (neoliberalism), they try to show that Nazism is against free enterprise and that the state intervenes in many aspects of this. Basically, they use a very US-centric view from late XX and XXI century to describe a movement from the interwar period. They start to highlight the "socialist" or "worker" terms in NSDAP's name, the party program in the early years, maybe some speech where some Nazi leaders use this term, Mussolini being a socialist pre-WW1 (in Italy's case) and so on.

This is wrong for a couple of reasons. As historians, we should be careful when applying the current understanding of an ideology or values to a situation in the past. As historians, we all know that conservative, liberal and socialist ideologies changed from the XIX century until today.

I will briefly stop on the welfare state and conservatism to make a point. The one that put the basis of the German welfare-state was Bismarck. He was, at that time, a conservative. Earlier he was in the Liberal party. He was never a socialist. By US standards of today, the Bismarckian laws for insurance are "socialist" when in reality Bismarck hated and fear socialism. He was never one. We also have Tory parliaments in XIX century UK that passed laws for regulating work in the factories. Romanian conservatives passed in 1912 one of the most modern worker insurance laws of the time. Conservatives in post WW2 Europe were instrumental in enacting the welfare states in the west. The view that state intervention in economy "smells" of socialism is new on a historical time scale.

After, hopefully, showing that the economic doctrine of ideologies are not static and sometimes are different in Europe and US, I will go on to the next point. The political and social context of the late XIX century and interwar Germany. This period sees an increased importance of mass politics and movements. Gone are the days of saloon politics of the 1850's Europe. The era of mass parties is coming and the SPD is the best example of this because it comes with the idea of modern membership (with a card) and with mass movement associations (trade unions, sports clubs, reading clubs etc). The Weimar Republic further democratizes the political sphere. Now the parliament has it's power increased and the president (which is elected) are up for grabs only to political parties and/or movements that are campaigning to the masses. SPD and Zentrum (through all the Catholic associations) are political powerhouses and the pre WW1 conservatives that did not try to go towards the masses are left on the sidelines. The Nazis are a "symptom" of this democratization. Hitler was well aware that in order to seize power, you need a party that appeals to as many people as possible. This is one of the reasons that they do not erase the "socialist" or "worker" from the name. Nazism (and fascism) are themselves products of this new environment. The old style elitist conservatism can not win elections and this is why they tried to back Hitler in 1933 as it had the votes.

All this being said, I think that now is time for a case by case analysis of some Nazi policies to see if they are really socialist or not.

1. A main pillar of Nazism is antisemitism. Is antisemitism a socialist policy? Of course not.
2. Nationalism and in NSDAP case racial superiority. Socialism in theory (in practice a little bit less) is internationalist and does not believe in the superiority of a race (in European context). We can also trace this nationalism through the Volkish ideas of pre WW1 Germany and Austria.
3. Nazism wants an Aryan society where class is of no importance. This is the opposite of the constant class struggle of socialism. A struggle that is also international.
4. Socialists have trade unions that are only for workers and the factory owners are not allowed to be members. The Nazis through DAF, as u/kieslowskifan wrote "tried to positioning itself as the replacement for worker-led unions and as an honest broker between manager and worker". This is the opposite of class struggle. A central point in socialism.
5. Feminism- socialists want equality between genders and are not opposed to women working in factories. The Nazis sees women mainly as homeworkers and far away from politics.
6. Private propriety and economic planning/state intervention. While true that NSDAP had a 4 year plan, it was for the purpose of rearmament. The did not nationalized industry or agriculture and state intervention was used by govts. in other countries after the Great Recession.

In closing, the Nazis are not socialists as their main ideological stances run contrary to most of the socialist ideas of the time. This idea really gained traction in the last 2-3 decades among US conservatives and does not use a serious historical analysis, but cherry picks some arguments to fit a narrative fit for a certain XXI century audience.

P.S.: I am once again sorry that I also touched some modern day politics in my post, but I feel that it is not possible to put this debate into a context without this.