The Spiegelungssatz for the Carlitz module

In this addendum to arXiv:1110.0292 we prove a function field analogue of the Spiegelungssatz. It provides a relation between the Galois module structure of the class group of a cyclotomic function field, and the Galois module structure of the class module introduced by the second author. The result is already implicit in earlier work of the authors, but not stated explicitly.