I started wondering about this for no particular reason I can think of. Who’s idea was it? Was it supposed to be symbolic of something? Did they just start passing out pamphlets at rallies titled: “Forget everything you THINK you know about public greetings!”?
The germans were told than it originated in the Roman Empire, and since they were the modern successors to the Roman Empire they would adopt the Roman salute
There is no evidence of it being used in Rome.
The German Nazis took the idea from their allies, the Italian fascists.
The United States was doing a similar thing with the Bellamy salute from the 1890s up until they weren’t buds with the Nazis any more.
There is art work going back to the 1700s depicting ancient Romans using the salute. Meanwhile, the idea has staying power: I recall that this salute appears in the 2005 HBO series Rome.
So it’s an urban legend among classicists that got picked up by several politicians in the early 20th century, and notably the Italian and German fascist movements.