The "socialism" part in the name of the party was just because it was a trendy term at the time. They could have been called something else if there was a different trend.

For example now the neo-Nazi party in Germany is called the National Democratic Party. There is also another far-right party (not as extreme, but founded by former Nazis nevertheless) called the Republican Party.

Anyways, the problem with the pre-WW2 Nazi Party in Germany is that it was a party which included many different wings. One of these wings was a left-wing called the Strasserites. The leaders here were the Strasser brothers, and they were the ones who drew up the 25 Point Program. However, by the early 1930's these guys had been expelled from the Nazi Party. One of the Strasser brothers went into exile in Czechoslovakia, while the other was murdered in the 1934 Night of the Long Knives, where most of the left-wing Nazis were killed off.

An interesting tidbit is that in the 1930's, the Nazi Party had Germany be one of the few countries in the world that carried out privatization of formerly state-owned companies. http://www.ub.edu/graap/nazi.pdf