Cells and Cacti

Let (W, S) be a Coxeter system, let phi be a weight function on S, and let Cact(W) denote the associated cactus group. Following an idea of I. Losev, we construct an action of CactW x CactW on W that has nice properties with respect to the partition of W into left, right, or two- sided cells (under some hypothesis, which hold for instance if phi is constant). It must be noted that the action depends heavily on phi.