The haircut to which I am referring has seen something of a resurgence in popularity today, but is often likened to popular hairstyles of the Nazi regime in the 1930s and 1940s. It is (usually) a men's haircut, sides and back shaved, with the top left long and swept either to one side with a very heavy part line, or swept straight back, heavily treated with pomade or oil in both cases.
A couple of examples (note, the men in these photos may have absolutely nothing to do with right wing movements at all, they just have a similar haircut to the one I am referring):
Here is actor Ralph Fiennes sporting the haircut while playing an SS officer in the period film Schindler's List.
And here is actor Brad Pitt wearing his hair in this style for the period film Fury.
Was this haircut actually a part of "Nazi haute couture," or was it adopted by them from somewhere else, or is it merely a fantasy of modern movies and TV that has grown popular on its own?
It is certainly not a Nazi original, but they did wear it. As you have noted, the 'undercut' style is distinguished by a very sharp transition from shaved sides to longer on top in contrast to a blended 'regular cut'.
The undercut requires less skill and is therefore cheaper and may simply have been common among the impoverished without acess to a skilled barber, of which there were many during the depression, which was a global phenomenon.
This article suggests the style was indeed popular amongst Germans during the 1930's, but also asserts its popularity amongst frenchmen as well (refering to the works of Robert Doisneau during the 30's). Its absence during the postwar period I would probably link that to the return of prosperity, and secondarily to association with Nazism. As for the return, the previous article links it to "the current [2011] heritage trend".