Camels +5 CS against cavalry.
This one's obvious. Horses are typically terrified of Camels in the real world unless they're raised with them. This is something Arabian and Persian generals used to great affect against enemies to their north.
Access to Truffles provide a 10% combat bonus against Elephant units.
Slightly less well known, War Pigs were often used against elephants in battle. Elephants are terrified of both pigs and fire, so it was common to light domesticated pigs on fire and send them into a group of war elephants. They would panic and trample their allies.
Just found this Thread, and Kudos to starting an interesting Topic, but I've got to modify your first post, because I happen to have the references in front of me!
The use of the 'war pigs' or flaming pigs happened exactly once: Battle of Beneventum, 275 BCE, Romans versus Pyrrhos of Epirus, and what actually won the battle was not a Charge of the Porcine Lit Brigade but elephants that got wounded, panicked, and stampeded back through Pyrrhos' lines, trampling his own infantry - something that did, in fact, happen frequently whenever Elephants were used in battle..
On the other hand, if horses are scared of Camels, they are freaking terrified of Elephants - to the horsey mind, anything that big should not be moving, and any group of horses (like a cavalry unit) goes into Instant Disorder near elephants - unless the horses have been raised with elephants and are used to them.
So,
Elephant Rules:
ALL horse-mounted units (Chariots, Light Cavalry, Cavalry, Knights, etc) have a -10 Combat Strength if next to an enemy Elephant Unit.
IF there is Ivory within the city radius where you raise a Horse Mounted Unit, that unit is Immune to this effect
Any Elephant Unit that takes Damage from Any Source, there is a 33% chance that the Elephants Panic and make an immediate attack on Any Adjacent Unit at the Elephants' Unmodified Combat Strength, minus 7 (presuming that the Elephants' crews will not attack).
And an Addition:
The Aztec Effect:
Any Civ that has no access to Horses through Resource or Trade, the first turn that any of their units are attacked by a Horse-Mounted Unit, those units have their Combat Strength Halved.