When Aragorn floats up on the river bank, his elven cloak is fastened only on his left shoulder because he was fighting with a sword when he fell into the river. His cloak stays on his left shoulder throughout getting on Brego, and the ride to Helm's Deep.(Yay!!!!) But, as he rides up to the gate of Helm's Deep, his cloak is fastened only on the right shoulder. (Arghhhhh!!!!) There would have been no reason or occasion for him to switch the cloak because a)he was just about to fall off of Brego anyway; b)if he was going to take the time to "fix" his cloak he would have fastened it in the front, not on the complete wrong side.