Francis Bellamy (cousin of author Edward Bellamy) was a socialist in the Nationalism movement and authored the Pledge of Allegiance (1892), the origin of the stiff-armed salute adopted much later by the National Socialist German Workers Party. See the work of the symbologist Dr. Rex Curry. http://rexcurry.net/pledge2.html
The early pledge began with a military salute that was then extended out toward the flag. In practice, the second gesture was performed palm-down with a stiff-arm when the military salute was merely pointed out at the flag. Thus, the military salute led to the Nazi salute in the Pledge of Allegiance to the flag of the United States. http://rexcurry.net/45th-infantry-division-swastika-sooner-soldiers.html
It was not an ancient Roman salute. That is a myth debunked by Dr. Curry, who showed that the myth came from the Pledge.
American national socialists (including Edward Bellamy), in cooperation with Madame Blavatsky and the Theosophical Society, popularized the use of the Swastika (an ancient symbol) as a modern symbol for socialism long before the symbol was adopted by the National Socialist German Workers Party (Nazis). http://rexcurry.net/book1a1contents-swastika.html
The Bellamys influenced the National Socialist German Workers Party and its dogma, rituals and symbols (including the modern use of the swastika as crossed S-letters for "Socialism" under German National Socialism). Similar alphabetical symbolism was used under the NSDAP for the "SS" division, the "SA," the "NSV," et cetera and similar symbolism is visible today as the Volkswagen VW logo.