I cannot say where and when this cliché evolved, but I want to try and make it the stereotypical physique a little more comprehensible by pointing out the specific organizational context of the Gestapo. The Gestapo already existed before 1933 as a state policy responsible for fighting crime against the state, i. e. subversion, terrorism and the like. Under National Socialism, of course, this fight was radicalized, as "the state" became a very narrow objective, "cleansed" of democratic tendencies, pacifism and Jews. Because it became very easy to be considered and enemy of the state, the Gestapo quickly had their hands full, oppressing political and "racial" enemies. However, while the Gestapo quickly came under influence of the NSDAP and the SS, it still was subordinate to the interior ministry, and as such, it was part of public authorities. In 1936, Himmler became chef of the SS and the German Police, which stayed separate, and in 1939, the Gestapo was incorporated in the Reichssicherheitshauptamt under Heydrich.

Despite the new ideological "challenges" after 1933, the Gestapo men were first and foremost experts in criminal investigation. The stereotype of the haggard spectacle wearer can be seen as a reflection of this perception as a bureaucratic type of perpetrator, in contrast to the uniformed SS, which is perceived as more soldierly. The Gestapo officer did not have to rely on physical strength, but on pen and paper. This, of course, was part of what gives us shivers when thinking about these men, who Hannah Arendt fittingly called "desk murderers."

The popular portrayal of Gestapo men in uniform leather clothing is more puzzling, because, like today, criminal investigators did not wear uniform at all, and there also was no informal "dress code" we know of. Real leather coats most likely do not depict reality, as these were neither light or comfortable nor would they have been affordable for most Gestapo members. There was, however, a cheaper, lighter alternative called the "Klepper coat", made of rubber coated cotton, and these coats seem to have been popular in Wehrmacht, German State Railways and Gestapo. It is possible that Gestapo officers appreciated this kind of pseudo-uniform, as leather coats were traditionally worn by high-ranking army officers and provided some military flair to the otherwise un-military Gestapo. In recent depictions, it may just be a way to ridicule undercover policemen who consider themselves incognito, but fail to blend in by wearing unusual "civilian" clothing.

I would love to hear if someone could pin down the emergence of this stereotype.