It is a classical result from universal algebra that the notions of polymorphisms and invariants provide a Galois connection between suitably closed classes (clones) of finitary operations $$f:B^n\rightarrow B$$ f:Bn→B , and classes (coclones) of relations $$r\subseteq B^k$$ r⊆Bk . We will present a generalization of this duality to classes of (multi-valued, partial) functions $$f:B^n\rightarrow B^m$$ f:Bn→Bm , employing invariants valued in partially ordered monoids instead of relations. In particular, our set-up encompasses the case of permutations $$f:B^n\rightarrow B^n$$ f:Bn→Bn , motivated by problems in reversible computing.