Economically speaking German fascism was a bit of mongrel. Certain elements were centrally planned and government run but capitalist markets operated, and many international corporations (including American ones) operated freely within Germany throughout Hitler's reign.

Nazi Germany was actually very popular with big business. Unions were banned, the government supplied cheap labour and later on, slave labour (jews mostly).

Before 1933 the Nazi members of the Reichstag aligned themselves with the left on socio-economic issues e.g. during 1929-1930 at the beginning of the Depression they defended workers' unemployment benefits against cuts. They also supported strikes. Their propaganda was anti-capitalist and officially (in the 25 point programme) they committed themselves to various anti-capitalist measures such as the abolition of 'interest slavery'. In Mein Kampf Hitler stated:

The development of Germany was much too clear in my eyes for me not to know that the hardest battle would have to be fought, not against hostile nations, but against international capital.

However once in power, most of the socialist promises were not implemented. Although the Nazis did implement

...price and wage controls, tight limitations on the use of profits, high taxes, increased government spending, stringent foreign exchange controls, rationing of raw materials, and imperious bureaucratic intervention in the private sphere that made the Zwangswirtschaft of World War I and the immediate post-war period seem trivial by comparison. (Turner, p. 338)

Germany's economic elite found that once their profits went up again they had little trouble accommodating themselves to Hitler's regime. The 'socialism' of National Socialism was distinctly limited. The Nazis respected private property (except for Jewish businesses) and they abolished independent trade unions, conscious that they needed businessmen's economic expertise. Their economic policies were designed to place Germany as securely as possible for the coming war and were masterminded by Hjalmar Schacht, who was not a Nazi.

Hitler admired Stalin and his Five Year Plans and said it was inferior only to the Nazis' Four Year Plan. He thought the capitalist West (especially Britain and America) were plutocracies run by Jews. However he also once remarked in private that he would not fleece his own people based on a 'Jewish' theory of exploitation (i.e. Marxism).

What the Nazis would have done if they had won the war is a 'what if'. Hitler never abandoned his aim of transforming Germany's postwar economic structure in order to allow a new meritocratic elite to emerge. In private he said it was a flaw to permit too much wealth to accumulate in too few hands (often 'undeserving' hands). The economy's future was adjourned until Germany's victory, when it could freely decide what to do. Henry Ashby Turner Jr contends that "to the end Adolf Hitler held to his quest for a "third way" between capitalism and socialism." (Turner, p. 338)

Nazism was completely opposed to the theory of socialism that promoted internationalism, the brotherhood of man and international class solidarity. This was absolutely anathema to them. They advocated the opposite: racial solidarity and racial struggle. Whereas socialists said the enemy was the capitalist class in all countries, the Nazis said the enemy were the Jews and Slavs (although their propaganda against the Jews had an anti-capitalist flavour to it).

Source: Henry Ashby Turner, Jr., German Big Business and the Rise of Hitler (New York: Oxford University Press, 1985).

I think that the 'Socialism' in 'National Socialism' reflects a few things:

Firstly, remember the NSDAP's past - it was formed as a merger of the National Socialists and the German Worker's Party, when both were just two of a very large number of tiny, more-or-less irrelevent parties in 1920s Germany.

This meant that the party was something of a broad church for a long time. Hitler was not always at the head of the party - he struggled to maintain control, and it was often less than clear what the party stood for. Gregor Strasser, assassinated during the Night Of The Long Knives in 1934 (though he had long since been politically outmaneuvered) represented a great threat to Hitler's control, by representing the more leftist elements of the party.

The somewhat vagueness of Hitler's precise ideas, combined with the broadness of the coalition that made up the Nazi party, is at least partly responsible for the confusion.

On the other hand, the term was quite deliberately intended to contrast with Marxian Socialism. You are correct in saying that the Nazis in general and Hitler in particular were rabid anti-communists. I think that National 'Socialism' was reflected in Hitler's plans for the volksgemeinschaft ("people's community"), intended to unite Germans despite traditional religious, regional and class differences, thanks to their 'shared blood'.

There was great stress placed on the idea that Germany must honour its workers as the foundation of it's return to greatness. It could be considered part of the Nazi's propaganda campaign that the term 'socialist' persisted in the name of the party - much of the vote they gained in the 1930s, when they transitioned from fringe party to the biggest in Germany's history, was from dissatisfied workers who had lost faith in the Weimar Government, at which the Socialist Party was often the core. The Nazis promised to do away with the constant bickering and maneuvering of political parties in the Reichstag and help people get jobs and the economy on track, an appealing proposition after 20 years of lurching from one financial crisis to the next.