I had to Google it, it tries to pass off as an Austrian economics institute but is really a Libertarian think tank in Alabama
FYI, the "Austrian school" of economics is probably the most libertarian type. Economists in that school tend to have fringe political commitments to extreme forms of libertarianism, e.g. the position that the state itself is a kind of illegitimate monopoly and should be replaced by private corporations fulfilling state functions.
As to the topic of the thread - there are two basic ways to view the question of right-wing and left-wing politics. One view is that right-wing and left-wing are largely matters of tactics, aesthetic, etc. and therefore contingent and entirely context-dependent. In this view is it obvious that the Nazis were extreme right as they were considered right-wing in the context of 1920s and 30s Germany, they worked in coalition with the other parties of the right and were opposed by the parties of the left and center. As soon as they got into government they acted remarkably quickly to completely destroy the traditional centers of leftist political organizing (the trade unions, the SPD, and the KPD) through the mass incarceration of these groups in concentration camps.
The other view is that left and right are sort of transhistorical categories with a fixed meaning whose application varies according to context. In this view, also, the Nazis were extreme right. Their political program was meant to place what they believed was racial inequality at the center of social life. The most fundamental distinction between right and left is egalitarianism vs aristocracy, or the belief that people are fundamentally equal against the belief that some people are intrinsically superior to others. With this in mind we can see that Nazi policies were designed systematically to produce and intensify inequality between the inferiors and their superiors, even to the point of exterminating those deemed sufficiently inferior.
The right-libertarians like to claim the Nazis were left-wing but this is largely due to two reasons, 1) they don't want to be associated with Nazis at all, and prefer to tar their political opponents on the left with that brush and 2) their ontological commitments lead to this conclusion anyway. Alvarez has helpfully explained why this is, the right-libertarians believe that human activity can be divided into 'public/state/government' and 'private/market' spheres. Activity in the state sphere is by definition coercive and not-free, while activity in the private sphere is by definition non-coercive and free.
With such a cartoonishly simplistic view of human society it is no wonder that they try to argue absurd positions like the Nazis were left-wing.
I should also note that as Ajidica noted, there were some elements within the Nazi Party that could be considered vaguely left-wing, at least relative to the rest of the Nazi party and to the other far-right parties such as the monarchist DNVP. The Strasserist wing of the party had its main base of support in the
Sturmabteilung, the street-fighting organization which had descended from the post-WW1
Freikorps, reactionary militias which were constituted to put down the German Revolution. This wing of the Nazi Party had all the beliefs about racial inferiors and superiors I mentioned above, but believed in a genuine class revolution, most importantly a replacement of the aristocratic officer corps-dominated Army with a "People's Army," basically, a renamed SA.
Of course, as soon as he got into power Hitler was having none of this and so he had Röhm, Strasser (Gregor, by this time Otto had already been exiled and later managed to escape Europe completely until after the war), and their important supporters murdered in the Night of the Long Knives, as a signal to Germany's traditional elites that there would be no social revolution under the Nazis.