I'm not quite sure a right as a rein on a horse is the correct path to follow. Most riders I know tend to leave the reins somewhat in a curve towards the horse's mouth. They only tug and tighten them straight (or right) when bringing a horse about or halting him - and even then, in a very quick fashion. When the reins are tight the horse keeps his head down and, on joy rides at least, this is not what the rider wants. I may be wrong about competition riding though.