The White Tiger is one of the four spiritual beings of Chinese mythology. It is sometimes called the White Tiger of the West, and it represents the west, the autumn season, the color white and the virtue of righteousness.
In Japanese, the White Tiger is called Byakko. To the Chinese, the tiger was viewed as the king of all animals and lord of the mountains. The male tiger was the god of war, assisting the armies of the Chinese emperors and demons that threatened the dead in their graves.
Fittingly, Byakko in FFXI is spawned using a Gem of the West and an Autumnstone.
Sources: Japanese Buddhist Statuary, "China and Japan Myths and Legends," by Donald A. Mackenzie.