We establish the existence of an optimal partition for the Yamabe equation in RN made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to -infinity, the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in RN that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.