The whole breaths on one side thing was not just for lances. In period men-at-arms were 'highly encouraged' (read all but forced) to fight right-handed. You wouldn't want to be sinister after all, so you were taught to fight dexterous! Dexter being the Latin for 'right' and sinister being the Latin for 'left.'
It was important so that people wouldn't be in each other's way when fighting in a line. Imagine a line of men with poleaxes or swords and everyone fighting with different handedness. It would be like a chaotic version of sitting close to someone at a crowded dinner table when the person immediately to your right is left-handed. You'd keep interfering with each other's ability to fight effectively. So not just lances, but most strikes in general were coming from the opponent's right and thus to your left. So you had to decide if you wanted stronger protection on the left, or better breathing and vision. It was a compromise. There are quite a few examples of both styles of breaths, but there are no examples I can think of that only have holes on the left side.