I was talking about this with my friend and she was like, "I heard it's just a Roman Salute." Be ready to be surprised at how many people will defend this crap.
So I think this is a fascinating odd corner of history.
The "Roman" salute is definitely not Roman, and it's not of fascist invention either. It's an urban legend among the classical studies people: a made-up non-fact that circulated around the fringes of classical scholarship for a couple hundred years.