In this paper we address the problem of approximating symmetric systems with systems with the same symmetry. We show that for periodic systems, a reduced order periodic system can be obtained by SVD-techniques. We also show that pointwise symmetries of the impulse response are retained after balanced model reduction. Both results are based on the fact that under certain conditions the SVD-reduction of a matrix with unitary symmetries leads to a lower rank matrix with the same symmetries. The results are applied to model reduction of an interconnected system.