Not sure if this is particularly a good enough answer, and i'm sure there are way better people to answer this, but I'll give it a go:

Really, there is no simple, objective way to answer this as many people will draw completely different opinions on the information we have. As you said yourself, some people will claim they were left, some people will claim they were right. Modern definitions of left and right wing politics are very broad and difficult to associate with Historical politics.

The Nazi party was known formerly as the 'National Socialist German Workers' Party'. The term Nazi was given to them by German media.

The name alone is fairly contradicting, you see, Nationalism is right wing politics, and Socialism is left wing. In their 25 point scheme they had both right wing and left wing ideals. In their own eyes, they were centralist. They needed backing from the industries that would want right-wing policies, and they needed backing from 'the people', whom generally want left wing policies. It has been argued that the Nazi's only made themselves appear left-wing or socialist in order to take votes and support away from the Communists (a form of left wing extremism).

In practice the Nazi were very right wing, they were fascists. Fascists of course being known for being right wing, but combining some left wing policies. The Nazi party had contradicting policies, which you can read here - http://www.historyplace.com/worldwar2/riseofhitler/25points.htm

In general though its said that they would drop left wing policies whenever they contradicted with their right wing agenda.

So to answer your question, both. Much like communism, they were extremist politicians that took ideas from the entire political spectrum. In my subjective opinion, they were more of a unique right wing party whom simply picked and chose some left wing policies that suited their idea of a greater German community.

I would love someone to expand upon my very friend explination.