If we want to get into myth making, maybe at one point before the schism the Jedi were a trinary order that reflected the Father, the Son, & the Daughter (whom we know they had contact with); Jedi-Bendu Shamans, Jedi-Ashla Guardians and Jedi-Bogan Knights. It's a neat way to recall Lucas's original idea about Yoda and The Emperor not being Knights like Obi Wan and Vader, but a sort of guru-sorcerer priesthood class that just trains the Knights and doesn't get into the whole martial, lightsaber wielding side of things personally.
You could even do a circle of time type myth and have the Sith schism play out as we saw in the Mortis Arc; The Knights of Bogan grow sick of the status quo and make a powerplay to remove the Bendu Priesthood from power and wind up destroying the Ashla Guardians (with whom they sought to share power) in the process. As the Knights of Bogan flee in rage and despair, while what's was left of the Jedi-Bendu and Ashla-Guardians would found a new order of Jedi Knights to fill the void and guard against the Return of the Sith.