A definition of Socialism:
a theory or system of social organization that advocates the vesting of the ownership and control of the means of production and distribution, of capital, land, etc., in the community as a whole.
If you take the point 7 of the 25 points of Nazism, you have this:
7. We demand that the State shall above all undertake to ensure that every citizen shall have the possibility of living decently and earning a livelihood. If it should not be possible to feed the whole population, then aliens (non-citizens) must be expelled from the Reich.
Other points give support to the mentioned:
13. We demand the nationalization of all trusts.
14. We demand profit-sharing in large industries.
18. We demand that ruthless war be waged against those who work to the injury of the common welfare. Traitors, usurers, profiteers, etc., are to be punished with death, regardless of creed or race.
The Nazi theory says that it's ultimate objective would be the German race controlling the rich lands of what once was the USSR, living in communities were comradeship is higly encouraged, work is higly praised, as well as sports, and there are good standards of living for everyone. Of course that there would be opponents who would not be tolerated, but you would have each new generation more with the National Socialist ideal in mind. Then, as the Nazis and Soviets belived, a "new man" would appear, one that would be in harmony with it's respective system. There were differences between the systems, of course, but in both cases we have humans living in communities were common good is the ultimate objective.
As for the defintion of the Nazis in the political spectrum, I will say that you can call Pinochet a right wing leader, as far as the popular notion of right wing economics is concerned - but the Nazis definately don't fit in the classical left/right spectrum (which I already mention to found outdated, because it makes a generalization of moral values and economic system, which do not need to always follow the same line). The Nazi ideal was not about maintain private ownership of the means of production, while the workers were just tools of the bourgeoisie who only seeks profity (this is what Marxist Socialists say about Nazism). Actually, in Nazism, it was the government who controlled the bourgeoisie, not the opposite. I would simply label Nazism as a collectivist regime were the German race has to cooperate and expand in order to achive the highest possible human development.