There are tons of interesting examples of duality in mathematics (Poincaré duality, Verdier duality, Stone duality, s-duality, Tannaka duality, Koszul duality, Spanier-Whitehead duality ... ). What examples are there of triality in mathematics?
Note: this is not a duplicate of the question about trichotomies in mathematics. A trichotomy is any sort of classification into three. A triality is a classification into three where the relationship between those three is some sort of equivalence relation (especially one emphasizing that though things appear opposite they are in some sense the same).