Guess what? Contrary to popular revisionism, the NAZI's were really pagan, humanist, socialist, environmentalist, elitist, totalitarian, of course anti-semitic and therefore leftist: "National Socialist Worker's Party" (Nationalsozialistische Deutsche Arbeiterpartei). This is why Stalin, Chamberlain and Roosevelt got along so well with Hitler for so long.
I am well aware of the NAZI party's socialist leanings and the current accepted definitions in FReeperdom. I too also, for the most part agree with them. However, IMHO its not as clear cut as you suggest. For example: the Fascist propensity for the inclusion of Capitalist elements (Industrial and Banking) in their political philosophy don't make them true Leftists/Communists. Remember Shindler of "Schindler's List" fame, and other German business owners, using concentration camp Jews to enrich themselves? The Fascists incorporated elements of Capitalism and used them as a 'means to an end' resulting in the collusion of a Centrally-run politcal and economic system (Left) and business (Right). IMHO this is how China has slipped easily from a Communist (Leftist) society to a Fascist one, which it is today.
True Leftists usually killed anyone outright that was involved in any capitalist endeavor and ownership of property (Mao, Stalin, Pol Pot, Kim Jung-Il, Castro). The Right coopted, colluded and incorporated them.
I agree that it seems that Fascists have many characteristics of classical Leftism, but the incorporation of elements of Capitalism seem to put the Fascists in a special catagory of the larger grouping called 'Totalitarianism'. Lucas in Star Wars seems to deal with and point to Fascism as the methodoolgy by which a Free, Democratic society can degnerate into Totalitarianism.
JMHO