In the present article, different types of semantics for the logic N4, the paraconsistent variant of Nelson's constructive logic with strong negation, will be considered. N4 will be characterized in terms of so-called Fidel-structures. It will be stated that Fidel-structures are equivalent to twist-structures. Further, the N4-lattices, generalization of N-lattices, will be defined and it will be proved that they can be represented as twist-structures. It will be proved that N4-lattices form a variety νN4 and there is a natural dual isomorphism between the lattice of subvarieties of νN4 and the lattice of N4-extensions.