We discuss Lagrangian and Hamiltonian field theories that are invariant under
a symmetry group. We apply the polysymplectic reduction theorem for both types
of field equations and we investigate aspects of the corresponding
reconstruction process. We identify the polysymplectic structures that lie at
the basis of cotangent bundle reduction and Routh reduction in this setting and
we relate them by means of the Routhian function and its associated Legendre
transformation. We end the paper with examples that illustrate the
applicability of our results.