In their seminal paper [1], Orda, Rom and Shimkin have already studied fully symmetric routing games, i.e. games in which all players have the same sources, destinations, demands and costs. They established the uniqueness of an equilibrium in these games. We extend their result to weaker forms of symmetry, which does not require a common source or destination. Considering routing games, we provide conditions under which whenever there is some symmetry between some players, then any equilibrium necessarily has these symmetry property as well. We then extend the symmetry result to general games.