Mechanical metamaterials based on Euclidean polygonal tessellations represent a new class of architectured materials with the potential to exhibit a wide range of mechanical properties. In this work, we investigate a new class of systems based on the generic hexagonal tessellation with trigonal rotational symmetry and show how this tessellation has the potential to exhibit a wide range of Poisson's ratios, including auxeticity, as well as a large spectrum of Young's moduli whilst retaining transverse isotropy. The tessellation was characterized through geometric expressions in order to identify which combination of geometric parameters lead to realizable, concave or convex configurations and Finite Element simulations were used to evaluate the mechanical properties of these tessellations. Furthermore, three additively-manufactured prototypes, representative of the entire Poisson's ratio range (i.e. negative, zero and positive Poisson's ratio) were experimentally tested and analysed using Digital Image Correlation. The results obtained from both simulation and experimental approaches demonstrate the mechanical capabilities of these tessellations and indicate how new auxetic metamaterials may be found by exploring the vast design space afforded by Euclidean polygonal tilings.