Let alpha be an endomorphism of an arbitrary ring R with identity and let M be a right R-module. We introduce the notion of alpha-symmetric modules as a generalization of alpha-reduced modules. A module M is called alpha-symmetric if, for any m is an element of M and any a,b is an element of R, mab = 0 implies mba = 0; ma = 0 if and only if m alpha(a) = 0. We show that the class of alpha-symmetric modules lies strictly between classes of alpha-reduced modules and a-semicommutative modules. We study characterizations of alpha-symmetric modules and their related properties including module extensions. For a rigid module M, M is alpha-reduced if and only if M is alpha-symmetric. For a module M, it is proved that M[x](R[x]) is alpha-symmetric if and only if M[x,x (l)]R[x,x(-l)] is alpha-symmetric.