Koszul duality for Coxeter groups

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 Beĭlinson–Ginzburg–Soergel [8] and Bezrukavnikov–Yun [9] in a geometric context, and of the first author with Achar, Makisumi and Williamson [4]. As an application, we show that the combinatorics of the "tilting perverse sheaves" considered in [6] is encoded in the combinatorics of the canonical basis of the Hecke algebra of (W,S) attached to the dual realization.