We consider a class of reductive linear groups defined in terms of weighted oriented graphs of a special sort that we call signed quivers. Each of these yields a symmetric quiver, that is, a quiver endowed with an involutive anti-automorphism together with signs for the vertices and arrows fixed by the involution. The orbits of the groups can be described in terms of the indecomposable symmetric representations of symmetric quivers. We provide a general description of the indecomposable symmetric representations and prove that their dimensions in the finite and tame cases constitute root systems corresponding to certain symmetrizable generalized Cartan matrices.