The flag is only reversed on the right side of the uniform.
Army Regulation 670-1, Wear and Appearance of Army Uniforms and Insignia.
Regulation 670-1 states that when authorized for application to the proper uniform the American flag patch is to be worn, right or left shoulder, so that the star field faces forward, or to the flags own right. When worn in this manner, the flag is facing to the observers right, and gives the effect of the flag flying in the breeze as the wearer moves forward. The appropriate replica for the right shoulder sleeve is identified as the reverse side flag.