Duad-syntheme duality on six points

A bijection from the points of one six-set to the totals of another extends by incidence to bijections from duads to synthemes, synthemes to duads, and totals to points. A duad maps to the intersection of its two point-totals. The same duality exchanges internal duads of a three-versus-three partition with cross synthemes of its corresponding partition. The latter rule follows by splitting the six cross perfect matchings into parity classes of three, yielding two triangles on the six totals.