Was fascism under the Nazis a left or right-wing movement?

Some might assume there is nothing to discuss in this question, but far-right elements in both US and international politics beg to differ. They insist that because the Nazi Party had the word “Socialist” in its name, and because Fascism grew out of a trade unionist movement in Italy around the end of the 19th century, Nazism is thus a left-wing movement.

Should you doubt that there is such a controversy, take a look at this Q&A thread on Yahoo! Questions. It presents arguments on both sides. Also, in its section on Fascism’s positioning within the political spectrum, the Wikipedia article on Fascism does it’s best to give a fair accounting of both sides. So, left-wing or right?