I am a Paramedic and recently at work I was told that there is a specific siren tone (a high pitched nose and then a low pitched noise) that is considered offensive in Jewish communities because it was used by the Gestapo / Nazi Criminals during the holocaust. Throughout my career I've heard "we don't use that particular tone" many times, but I was always just under the impression that it wasn't well liked. No specific reason was given for it. It wasn't until yesterday that an older, Jewish Paramedic mentioned it to me that I heard this explanation. Another medic chimed in saying he was surprised that I did not know this about that siren tone. Only a couple of our ambulances even have this tone and sometimes I would use it for novelty, not aware that it was considered offensive / insensitive. Sometimes using a different tone gets the attention of drivers better. I simply couldn't believe that I'd never heard this before so I took to the internet.
I did a few google searches but those returned no results. I was able to dig up information on the Stuka Sirens but this tone is not similar to that at all. I did dig up this strange page that makes reference to a "nazi siren" but it is useless as a source.
I suspect the patient zero for this trope is George Stevens' 1959 adaptation of The Diary of Anne Frank which ends with the "siren" of an approaching Gestapo truck and heavy knocks on the door, before cutting to a forlorn Otto Frank in the present. (I've tried and failed to find a relevant clip online, sorry)
My suspicions - and feel free to treat them as such - that this is the trope originator, is that there are no references to specific Gestapo sirens in any of the narratives I've read, and the whole idea of the Gestapo even having a siren seems a tad counterintuitive to a shadowy state security agency.
To stick with Anne Frank (or more specifically, Anne Frank: The Book, the Life, the Afterlife by Francine Prose):
The car arrived without sirens, without haste [...] no one in the office below was alarmed by the appearance of the car until a fat man appeared, and, speaking in Dutch, ordered everyone to be quiet. One of the men who got out of the car wore the uniform of a sergeant in the Jewish Affairs Section of the Gestapo in Holland.
Anne Frank '59 is very well studied by film historians as it played a central role in the globalisation of Holocaust understanding. Along with the release of the Diary as a book and play earlier in the 1950s and the opening of the Anne Frank House in Amsterdam in 1960, it established a particular visual language for the Holocaust that permeated popular culture, including that specific siren sound.
Without understanding musical terminology in the slightest, your low note/high note blast is likely to be simply the common siren scheme found in Europe - ("nee naw nee naw" to children everywhere) - which is suitably discordant and alien enough for American filmmakers in the 1950s to adopt as shorthand for "The authorities are coming (but they're not like your authorities, they're Europeans)!"
Of course it's possible that some very specific memories have added to this where local police agencies collaborated with the Gestapo and survivors/emigrants came to associate this particular sound with the Nazis - understandable if your only encounters with the police came as a result of occupation - but my conclusion is that it's a piece of folklore.
But hey, associations are subjective so is perception any less valid than fact? Oh, it's r/AskHistorians, so yes.
I can say with a high degree of certainty that there was no specific Gestapo siren. There is no mention of such in the pertinent literature on the Gestapo or Carolyn Birdsall' book Nazi Soundscapes: Sound, Technology and Urban Space in Germany, 1933-1945. Additionally, it is not accurate to imagine the Gestapo as an organization similar to regular police.
The Gestapo, which is short for Geheime Staatspolizei, was the Nazi successor to the what in the Weimar Republic had been the political police, such as the Preußische Geheimpolizei. A political police is, as it sounds, a police force charged with the prevention of crimes of a political nature - something, which albeit it had a tradition e.g. in the German Empire had seem very essential during Weimar times because of the excessive and often violent nature of political conflict, for example between communist and right-wing groups.
The Gestapo was specifically charged with the persecution of political enemies, which in the Third Reich included Jews. In 1936 the regular police and the SS were merged and the regular police (e.g. the criminal police, the Schutzpolizei, which was the people patrroling the streets in uniform, the Gendarmerie, the rural uniformed police, and other branches of police forces) came under the control of Heinrich Himmler and the SS. While this transferred massive executive powers into the hands of the Gestapo and laid the ground work for them becoming a sort of judge-jury-executioner all in one insitution later on (during the war, the Gestapo could take over cases and hand out punishment imediately without a trial).
Still, the Gestapo is best not imagined as something resembling a regular police force with police cars with sirens. You couldn't get on your phone and call the German equivalent of 911 (which is 110 btw.) and get the Gestapo to come to some place with sirens blaring. Rather, they either came because people were suspected of being political enemies or to supervise the arrest and deportation of Jews e.g. Also, if they came to arrest a person, they did not do so using sirens and blue lights but rather turned up at your doorstep at 4am in the morning. It's best to imagine them akin to the homocide division in American police forces, i.e. they get called in when needed or set out to investigate and arrest people.
When they needed such a thing like a siren (like in the case of the Greiferkommandos, squds of police under Gestapo supervision roving the streets of Berlin in 1942/43 and arresting people wearing the Jewsih star at randomo), they used cars from the regular German police and their sirens might have sounded similar to what you have in mind.
All in all though, sirens are not - as far as my knowledge goes - particularly associated with the Gestapo and I too like /u/JDHoare suspect that this might be something that started as a pop culture trope rather than having a historical basis per se.