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!”?
As best as I can understand, some anonymous person in the early modern period, 1600s-1700s made up this idea that the ancient Romans did this funny salute. It’s an urban myth. There’s no evidence that real ancient Romans ever used this practice.
Anyhow, the idea floated around for a while; it shows up in paintings in the 1700s. It was picked up by several political movements in the early 20th century, including Bellamy in the United States, and Mussolini, who was a big Romaboo. The Nazis did get the idea from their Italian allies.