Think of it as placing a sticker on your arm. Obviously you have to rotate the sticker when moving it from arm to arm so the face-up side points out. The same concept works for textures.
In order to mirror an arm texture, you not only have to flip it over, you also have to displace the entire texture, that means dragging it 20px either to the left or the right, I don't remember which. The templates should tell you this.
