Rings of formal power series with exponents in a cyclically ordered group were defined in . Now, there exists a “valuation” on : for every in and in , we let be the first element of the support of which is greater than or equal to . Structures with such a valuation can be called cyclically valued rings. Others examples of cyclically valued rings are obtained by “twisting” the multiplication in . We prove that a cyclically valued ring is a subring of a power series ring with twisted multiplication if and only if there exist invertible monomials of every degree, and the support of every element is well-ordered. We also give a criterion for being isomorphic to a power series ring with twisted multiplication. Next, by the way of quotients of cyclic valuations, it follows that any power series ring with twisted multiplication is isomorphic to a , where is a subgroup of the cyclically ordered group of all roots of in the field of complex numbers, and , with a totally ordered group. We define a valuation which is closer to the usual valuations because, with the topology defined by , a cyclically valued ring is a topological ring if and only if and the cyclically ordered group is indeed a totally ordered one.