The work describes the manufacturing, testing and parametric analysis of cellular structures exhibiting zero Poisson’s ratio-type behaviour, together with zero and negative stiffness effects. The cellular structures are produced in flat panels and curved configurations, using a combination of rapid prototyping techniques and Kirigami (Origami and cutting) procedures for PEEK (Polyether Ether Ketone) thermoplastic composites. The curved cellular configurations show remarkable large deformation behaviours, with zero and negative stiffness regimes depending also on the strain rate applied. These unusual stiffness characteristics lead to a large increase of energy absorption during cyclic tests.