Power-Ups for Mounted Combat

[Varyon]

"Trick Riding" or a similar name could be a Technique that lets your Riding skill function as Acrobatics for purposes of enhancing defenses, or even for some types of acrobatic movement (evasion and spinning come to mind). I'd eyeball it at -4, thus [5] to get it up to full skill. Note this isn't a case of your mount using Acrobatics, it's more you expertly guiding them to help you avoid attacks, weave around foes, and similar.

Perks that grant Sacrificial Parry and Sacrificial Block are useful for defending your mount. For heavily-armored mounts, a Perk for both mount and rider might justify any successful hit on the rider allowing the mount to attempt a Dodge - on a success, the mount gets in the way of the attack and is struck instead in the Torso.

Cavalry Training is a great Technique with aggressive mounts, and for tramples, as it lets you strike along with your mount at no penalty. A foe who is lanced, body-checked, and then trampled is a foe who is having a very bad day.

Buying off the Horizontal penalty to kicking (as noted upthread) could be built as a Technique - for mounts with kicking damage below 2d, this is -2 to skill, while for those with 2d or higher kicking damage it is -4 to skill (-1/die to damage requires +1 to damage for less than 2d, +2 to damage otherwise, as +2 to damage can be rewritten as +1/die). That's either [3] or [5] to negate, +[1] if you require Unique Technique (I wouldn't). This is probably a Technique of the mount's, although you could justify having the rider with it instead. Alternatively, you could build it as a quasi-Advantage, using Natural Weapons (Pyramid #3/65*). Horizontal functionally includes Reduced Damage -30% for your crushing Natural Weapon, and as a Crushing Natural Weapon has a base cost of [5], that's [-1.5] to cost. We'll round up to cost [2], which is also the cost to negate that trait for cutting and impaling weapons, so it works for any mount (unless you have some weird mount that does corrosion or fatigue damage, in which case the trait costs [3]). This is a general Advantage for the mount; if it only works with a specific rider you could drop the cost to [1]. If the rider grants the ability, I'd probably charge [2] for it, or [1] if it only works with a specific mount.

For enhanced trampling, I'd allow the mount to use its full ST for determining damage. An Advantage that has a "half ST" effect is Extra Arm, where making said a limb have half ST is worth [-2.5]. Increased "Trampling ST" isn't quite as useful as increased Arm ST, so I'd make it [2], like the alternate build for increased kicking power above. Again, the mount or rider can purchase this for [2], or one can purchase it for [1] and only be able to use it with a specific mount/rider. Beyond that, Striking ST (Trampling Only -60%) [2]/level** is the way to go.

Striking ST (Mounted Only -40%?) [3]/level might be useful for a character who expects to fight from the saddle frequently. Add another -20% to the Limitation value, for [2]/level, if it only works in a charge.

Optionally, some (or all) of the above may have a prerequisite of Special Rapport, having the mount be the Ally of the character (or vice versa - this is GURPS, so an intelligent PC horse with a knight Ally isn't out of the question), or similar. This may be worth a reduced price on some of the traits.

*I have now recommended this article enough times I don't even need to look up which Pyramid number Alternate GURPS III is anymore.
**[1.5]/level for SM+1 mounts, [1]/level for SM+2 and larger mounts, thanks to reduced ST costs for large creatures.