We introduce a generalisation of Nelson algebras having a not necessarily involutive negation. We suggest to dub this class quasi-Nelson algebras, in analogy with quasi-De Morgan lattices, which are a non-involutive generalisation of De Morgan lattices due to Sankappanavar. We show that, similarly to the involutive case (and somewhat surprisingly), our new class of algebras can be equivalently presented as (1) quasi-Nelson residuated lattices, i.e. models of the well-known Full Lambek calculus with exchange and weakening, extended with the Nelson axiom; (2) non-involutive twist-structures, i.e. special binary products of Heyting algebras, which generalise a well-known construction for representing algebraic models of Nelson’s constructive logic with strong negation; (3) quasi-Nelson algebras, i.e. models of non-involutive Nelson logic, viewed as a conservative expansion of the negation-free fragment of intuitionistic logic; (4) the class of bounded commutative integral residuated lattices which satisfy a universal algebraic property that we introduced in a previous paper, called $(0, 1)$-congruence orderability. The equivalence of the four presentations, and in particular the extension of the twist-structure representation to the non-involutive case, is the main technical result of the paper. We hope, however, that the main impact will lie in the possibility of opening new ways to (1) obtain deeper insights into the distinguishing feature of Nelson’s logic (i.e. the Nelson axiom) and its algebraic counterpart (the Nelson identity); (2) be able to investigate certain purely algebraic properties (such as 3-potency and $(0, 1)$-congruence orderability) in a more general setting.