In this paper, we propose a new approach to compute the projection of a set defined by polynomial equations. It assumes that the polynomial equations have some nice symmetries and that a solution of the projection problem is already available in the case where the variables along which we project are all positive. A new interval-based algorithm which combines symmetry operators and set quotient is proposed. Symmetries are used to move from one part of the space to another. The set quotient is needed to avoid redundant symmetries. The projection procedure yields an inner and an outer approximations of the projected set. Two applications are considered. The first one corresponds to the characterization of the space occupied by a rotating polygon, and the second one deals with the estimation of the speed of a moving object observed by several robots with uncertain orientations.