Recent work by Busaniche, Galatos and Marcos introduced a very general twist construction, based on the notion of _conucleus_, which subsumes most existing approaches. In the present paper we extend this framework one step further, so as to allow us to construct and represent algebras which possess a negation that is not necessarily involutive. Our aim is to capture the main properties of the largest class that admits such a representation, as well as to be able to recover the well-known (...) cases—such as _(quasi-)Nelson algebras_ and _(quasi-)N4-lattices_—as particular instances of the general construction. We pursue two approaches, one that directly generalizes the classical Rasiowa construction for Nelson algebras, and an alternative one that allows us to study twist-algebras within the theory of residuated lattices. (shrink)