Cells and cacti

Let $(W,S)$ be a Coxeter system, let $\phi$ be a weight function on $S$ and let $\mathrm{C}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{\_}W$ denote the associated {\it cactus group}. Following an idea of I. Losev, we construct an action of ${$\mathrm{C}\mathrm{a}\mathrm{c}\mathrm{t}$}_W \times {$\mathrm{C}\mathrm{a}\mathrm{c}\mathrm{t}$}_W$on$W$ which 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 or if $W$ is finite of rank $\text{textless}5$). It must be noticed that the action depends heavily on $\phi$.