It was certainly a right-wing regime. German politics at the time had the following rough divisions:
A far left wing, personified by the KPD and USPD, which wanted a radical redistribution of wealth from rich profiteers to people at all levels; a social reorganization, doing away with noble, military, landowning, and industrial elite rulers; and an end to class distinctions and to military adventures that benefited those elites.
A center-left, mostly the SPD (mainstream socialists), which the USPD broke away from during World War One because the SPD wanted to support the war effort. The SPD wanted a more democratic and less autocratic progression to socialism, with a welfare state.
A center, which included the Zentrum (the Catholic Center Party, with mostly Catholic voters, advancing Catholic concerns) and the DDP (the German Democratic Party, an economically liberal ['liberal' in the economic sense, not in the American left-wing sense] party which received more Jewish votes than other parties did), and which tended more to social inclusion than the far right, but didn't want to completely do away with all class distinction or profit-making as the far left.
A center-right, best personified by the DVP, which accepted "traditional" social roles and capitalist profit-making and class distinctions of one kind or another, but which also was too "internationalist" (that is, they cooperated with France and America in renegotiating reparations payments) and not militaristic enough for the far right.
And a far right, personified by the DNVP and the Nazis, which absolutely rejected initiatives of international amity with the former Allies of World War One; which wanted a remilitarized Germany; which did away with many vestiges of class such as the nobility, and which asserted military primacy over the landowning and big bourgeois industrial classes, but which also allowed all the profiteering a fat-cat could want, as long as Hitler had no vendetta and the war effort wasn't hindered (the Nazis' original 25-point program, see http://avalon.law.yale.edu/imt/1708-ps.asp, contained points against war profiteering, but this was of course never enacted, in the case for instance of officials such as Goering, who enriched himself beyond belief); which hated trade unions; which wanted a return to "traditional" social roles (a curtailment of liberation of women or gay people); which was virulently nationalist and almost completely bigoted (though the DNVP did have Jewish founder members such as Fritz Arnold); which was reactionary against the Communist and Socialist revolts that sprang up across Germany especially in 1918-23; and which intended (in the Nazis) war to reclaim Germany's lost land in the east.
Briefly: on the left, an end to profiteers amassing huge wealth; to Germany's old military, landowning, industrial, and noble hierarchy; and to military imperialism. On the right, complete acceptance of profiteering (provided the war effort wasn't hindered); continuation of old hierarchies except the royalty and nobility, with the military at the top; nationalism that always sought a group to exclude, whether by ethnicity, national origin, or religion; and an almost insane devotion to military rearmament and conquest, and explicit embrace of imperialist, aggressive warmaking. Those were (and still are) right-wing traits, and those were the traits of the Nazis. They had very few initiatives that could be described as "socialist" except their nationalization of industries and materiel crucial to the war effort (and in that context, America and Britain did the same thing; see http://www.nytimes.com/2008/10/13/business/worldbusiness/13iht-nationalize.4.16915416.html?_r=0).
People on the American right have lately claimed that the word "socialist" in "National Socialist" means that the Nazis couldn't have been right-wing, but this is nonsense, as Hitler busted unions, sided with big bourgeois management, cut welfare payments (as Professor Richard Evans writes about in "The Third Reich in Power"), etc., and hardly enacted any socialist policies at all (unless someone thinks building roads makes a country "socialist").