The anisotropic elastic mechanical properties of a family of single material chiral mechanical metamaterials are explored systematically. An integrated monoclinic-micropolar constitutive model is developed to quantify the anisotropic mechanical properties of the chiral designs with different geometries. The model predictions are thoroughly verified by mechanical experiments on three-dimensional (3D) printed specimens and finite element simulations with periodic boundary conditions. The new integrated monoclinic-micropolar model can predict the anisotropic elastic properties in all directions. Normalized model parameters for this family of chiral designs are provided. Finally, the anisotropic effective stiffness and effective Poisson’s ratio of all geometric designs in this family are quantified. The anisotropy and the completeness of auxeticity are evaluated systematically.