Two simple involutions on the set of Dyck words will be considered. The first one corresponds to the mirror image of the path coded by the word, and the second one is related to the rank parameter on graphs introduced by Baker and Norine. The composition of these two involutions is a permutation which has an interesting behavior. Some conjectures and results on this behavior - obtained in collaboration with M. Barnabei and F. Bonetti - will be presented.