Gas Detection Brassard
Fearing possible poison gas attacks from the German defenders on D-Day, Allied soldiers were outfitted with gas masks and two gas detection brassards (armbands). Coated with a chemical, the paper brassard was designed to change color when exposed to certain gasses, thus warning a soldier that he was in potential danger.
The brassard, which was fragile and easily torn in combat, was intended to be worn on the left shoulder so as not to interfere with the firing of a weapon. One brassard was meant to be worn into combat, with the other safely stored along with the soldier's gas mask in a waterproof bag worn on the chest.
Both the brassards and gas masks would prove unnecessary as the Germans, themselves afraid of retaliatory gas attacks on their own troops and/or people, did not use such weapons.
Fact vs. Fiction
The medical officer shown above, as well as most of the actors in the film, wear their gas detection brassards on their right shoulder instead of their left shoulder.