Introduction of the stirrup

[Peroni]

Ah, but that's what the cataphract mount's chamfron and neck armour
are for. You can slash either side and not worry about your horse.

As for the household cavalry - or any any others not carrying shileds -
it really wouldn't matter which side they turned to the enemy. Though,
I guess it would actually pay them to keep the right side to the
enemy, wouldn't it? At least that way, they'd have a greater reach to
parry with their own swords against an enemy sword.

Only if you lost your primary weapon, your spear. I would still rather use a shield to protect myself than a blade!

The Chamfrons rarely covered the horses ears.