Lewis-Zagier correspondence for higher order forms

The Lewis-Zagier correspondence, which attaches period functions to Maa\ss\ wave forms, is extended to wave forms of higher order, which are higher invariants of the Fuchsian group in question. The key ingredient is an identification of Higher order invariants with ordinary invariants of unipotent twists. This makes it possible to apply standard methods of automorphic forms to higher order forms.