Saddle

Saddles are items only found in the chests found in map-generated underground dungeons; they cannot be crafted. They can be placed on pigs which will allow the player to ride them by right clicking the saddle. However, the pig is uncontrollable. Saddles cannot be placed on mobs other than pigs. The player can climb off and on the pig as they please. The saddle cannot be recovered from the pig and will be lost if the pig disappears from the map or is killed. The pig will take any fall damage that may be applicable.

This video demonstrates a player riding a pig (and being suffocated in a tree).