On symmetric quadrangulations

This note gathers observations on symmetric quadrangulations, with enumerative consequences. In the first part a new way of enumerating rooted simple quadrangulations is presented, based on two different quotient operations of symmetric simple quadrangulations. In the second part, based on results of Bouttier, Di Francesco and Guitter and on quotient and substitution operations, the series of three families of symmetric quadrangulations are computed, with control on the radius.