As it is not clearly stated here so far: As far as I know **there is still no conceptual explanation of this observation**.
We were looking at this situation for unitary reflection groups in Proposition 2.1 in our last year's preprint http://arxiv.org/abs/1404.5522, tried to give an explanation and discussed possible approaches with David Bessis. However, this lack of understanding still seems to be the state of the art, even for finite Coxeter groups.