Democrats and Republicans each like to accuse the other of being the real nazis or the real fascists. It's a very stupid and cowardly game that they play, trying to avoid a label or an association. It has everything to do with appearances and nothing to do with substance.
What Republicans Say
The whole question is confused by the fact that, in vulgar American usage, the meanings of conservative and right-wing have changed. Nowadays, to be conservative is to be an advocate of what used to be called liberalism. In most of the world, what is called conservatism in the USA -- the demand for free markets and less government -- is still called liberalism.*
According to this postwar pseudo-conservatism, any big, powerful government is ipso facto a leftist government.
A generally neglected implication of this redefinition of conservatism is that the powerful absolute monarchies that existed before the Enlightenment must be considered leftist governments. That is utterly absurd. This redefinition clearly was not well considered.
For Republican rhetoric, an important effect of the redefinition of conservatism is that it is now considered impossible to be right-wing and socialist at the same time. On that basis, Republicans can say that National-Socialism and Fascism are leftist and in no way conservative.
Before World War II, that was not the case.
Most Americans have never heard of Tory Socialism. Tory means Conservative. Tory Socialism means Conservative Socialism. Conservative socialism as an idea started in the UK, where it was originally associated with the Young England movement in the 1840s.
How can socialism be conservative? Conservative socialism means making concessions to the needs of the laboring class in order to recruit that class as defenders of established traditions and institutions.
The idea was implemented in Germany by Otto von Bismarck, who, after essentially banning the Socialist Workers' Party in 1878, created the German welfare-state in 1881. The purpose was to eliminate grievances that the Marxists could use to gain popular support. Bismarck did all this with support from Conservatives.
Both Mussolini and Hitler followed in Bismarck's path, doing what Bismarck had done but more of it. National-Socialism and Fascism can be regarded as left-right syntheses in regard to methods, but the ultimate aim is conservative.
It is only by relying on a very limited frame of reference that today's so-called conservatives can argue that National-Socialism and Fascism are in no way right-wing or conservative.
What Democrats Say
The Democrats, eager to throw the hot potato back to the Republicans, argue that National-Socialism and Fascism are not socialism. It's a purely semantic argument. They can't deny that Hitler and Mussolini created large-scale social programs. So, they point out that Hitler and Mussolini did not implement "government ownership of the means of production" the way Stalin did. But that is not a universally shared definition of socialism. That's just the definition that they want to use just for this particular argument. There are many socialist and social-democratic parties that do not insist on government-takeover of enterprises, whose claim to being socialists is for some reason not challenged.
One could argue that the reason why social-democrats in various countries do not nationalize industries is that they are kept in check by opposition. But this was also true for the Fascists in Italy and the National-Socialists in Germany. Yes, Hitler and Mussolini were autocrats, but it does not mean that they could disregard what everybody else thought.
When Mussolini established Fascism in Italy, it was with the consent of King Victor Emmanuel and the rest of the conservative Italian establishment. Under the circumstances, that establishment continued to wield influence and Mussolini could not implement everything in the Fascist agenda. But in the Salò Republic (1943-1945) the Italian Fascists actually did implement nationalization of every enterprise with more than 100 employees. That is rather socialist by anybody's definition.
The NSDAP in Germany also had to compromise with the conservative establishment, which is why figures on the left wing of the NSDAP became disgruntled and began talking about the need for a second revolution, and consequently had to be suppressed with the "Night of the Long Knives" in 1934.
Conservatism of Means vs. Conservatism of Ends
After the Second World War, the idea of using government for conservative purposes (as, for example, with anti-miscegenation laws, or restrictions on abortion and pornography) came under attack.
Liberalism thus became the new conservatism. It meant that certain ways of doing things -- the liberal ways -- had to be maintained. Under this pseudo-conservatism, the Constitution and the free market are not means to an end, but ends in themselves (much like some religious law). Those principles are treated as holy, regardless of whether the country is going to ruin because of them.
Another way to say it is, presentday American conservatism is a conservatism of means. As long as we keep doing certain things the same old way, the so-called conservatives can claim victory. All the leaders of the Soviet Union between Kruschev and Gorbachev could claim to be conservative in that very same sense, of refusing to adapt. This "conservatism" is in fact rigidity.
Fascism and National-Socialism by contrast represent conservatism of ends. Fascism and National-Socialism looked at their most cherished values, and chose means that they thought would conserve those values.
For Italian Fascism, the most cherished values were cultural, while for the NSDAP the most cherished value was the racial quality of the German people. More fundamentally stated, Italians were concerned with continuing to be Italians, and Germans were concerned with continuing to be Germans. The two nations chose means that were in some ways the same, in some ways different, for essentially the same purpose.
So, notwithstanding all the disingenuous blather from presentday American political parties, the bottom line is that National-Socialism and Fascism were simultaneously socialist and, in regard to the survival of their nations, conservative.
* The redefinition of conservatism had not fully taken hold in 1951. Anti-Communist commentator Upton Close was still defending the traditional definition of liberalism: "Liberalism is the system which gives the opportunity for people to earn their own way, and make their own successes or failures, to earn what they get, and get what they earn, and keep what they get. so long as it is not taking from others their opportunity to get and keep. Liberalism is the system whereby people have as little government, as few laws and officials over them as possible, taking as little of their savings and earnings as possible in taxes."(Upton Close, broadcast of 1 April 1951) Today this liberalism would be recognized as the ideology of Ronald Reagan and Rush Limbaugh.