  We construct a "Koszul duality" equivalence relating the (diagrammatic) Hecke
category attached to a Coxeter system and a given realization to the Hecke
category attached to the same Coxeter system and the dual realization. This
extends a construction of Beilinson-Ginzburg-Soergel and Bezrukavnikov-Yun in a
geometric context, and of the first author with Achar, Makisumi and Williamson.
As an application, we show that the combinatorics of the "tilting perverse
sheaves" considered in arXiv:1802.07651 is encoded in the combinatorics of the
canonical basis of the Hecke algebra of $(W,S)$ attached to the dual
realization.
