Why is the U.S. flag sewn on backwards on arms of military uniforms?
Yep. LookingAround has the link.
The basic idea (so you don’t have to read it) is that the flag is always “in motion”, relative to the wearer. On your left arm, as you move forward, it streams with the stars on the top left. On your right arm, it would stream with the stars on the top right.
The same convention applies to flags painted on vehicles, especially airplanes.
-butler
And to answer the unasked question of “Why not put on left sleeve and have flag look “correct”?”, the right shoulder is considered the place of honor, and the left had been the spot for unit insignia for the US Army.
And in the past they had placed it on the left sleeve at times, but it was a kind of ad-hoc thing: Until a couple of years back, only foreign-deployed troops actually wore the flag on their sleeves. Now they’ve made it a fixed feature of the BDU.
Left sleeve wears your current command-unit patch, right sleeve also wears the command-unit patch from your last war (thus you may see a vet of Desert Storm, activated to the current war, wearing a 65th RCOM patch on the left sleeve and a 1st Cav on the right).