  The forward and backward scattering off linear systems with discrete
rotational symmetries R_z (2{\pi} /n) with n $\ge$ 3 are shown to be restricted
by symmetry reasons. Along the symmetry axis, forward scattering can only be
helicity preserving and backward scattering can only be helicity flipping.
These restrictions do not exist for n < 3. If, in addition to the n $\ge$ 3
discrete rotational symmetry, the system has duality symmetry (obeys the
helicity conservation law), it will exhibit zero backscattering. The results
pinpoint the underlying symmetry reasons for some notable scattering properties
of R_z (2{\pi} /4) symmetric systems that have been reported in the
metamaterials and radar literature. Applications to planar metamaterials and
solar cells are briefly discussed.
