Mechanical metamaterials with zero or negative Poisson’s ratio were subject to increasing research interest over the last few years. Their energy absorption capabilities make them suitable for impact and dampening applications, such as personal protection equipment or packaging materials. The variable porosity and unusual mechanical properties also make them applicable in drug delivery systems and wound management. Herein, we present an extension to common auxetic structures, including tetra-chirals and tetra-antichirals. By introducing an asymmetry in the design of their unit cell, Poisson’s ratio can be varied over a broad range. Specimens with a selected amount of asymmetry were additively manufactured with a thermoplastic polyurethane using fused filament fabrication. Compression tests were performed to investigate the influence of the asymmetry on Poisson’s ratio and the compression modulus. Two different numerical models were employed using ABAQUS to describe the mechanical properties of the structures and were verified by the experiments. The numerical models are based on three-point bending test data. Both asymmetric designs show an influence of the asymmetry onto Poisson’s ratio, resulting in variable Poisson’s ratio, porosity, and compression modulus.