TITLE: Examples of triality in mathematics
QUESTION [15 upvotes]: 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).

REPLY [5 votes]: An interesting duality is polarity with respect to a conic in the projective plan. A conic is determined by a bilinear form $q$ (the equation of the conic being $q(u,u)=0$) and one way to describe the polarity is to say that the polar of a point corresponding to a vector $v$ in the underlying $3$-space is the line whose equation is $q(v,\cdot)=0$.
Now, this yields an immediate equivalent for trialities: consider a cubic, it is determined by a trilinear form $t(\cdot,\cdot,\cdot)$ and two points (with corresponding vectors $u$, $v$) and a line are in triality if $t(u,v,\cdot)=0$ is an equation of the line.