  Quantum states of angular momentum and spin generally are not invariant under
rotations of the reference frame. Therefore, they can be used as a resource of
relative orientation, which is encoded in the asymmetry of the state under
consideration. In this paper we introduce the analytical characterization of
the rotational information by parameterizing the group characteristic function
by polynomial functions. By doing so, we show that the set of states achievable
through $SU(2)$-covariant channels admits an analytical characterization and
can be studied through the use of semidefinite optimization techniques. We
demonstrate the developed methods via examples.
