Simple geometric distortions applied to the isometric Kelvin cell structures, (the tetrakaidecahedron), are shown to result in equivalent materials with anisotropic Hooke's tensors. The equivalent material models are estimated using a recently published inversion method where the 21 independent elastic constants of the Hooke's tensor are identified. In these cell geometries, some of the faces of the Kelvin cell have been twisted and/or tilted. Numerical experiments suggest that the equivalent material models of the distorted cells exhibit variations in compression, shearing, shear-compression and shear-shear coupling moduli, which are shown to be continuous functions of the degree of twist and tilt applied. When twist and tilt are combined, it is demonstrated that full anisotropy in the elastic properties may be generated. A rotational symmetry without symmetry planes, but having either a tetragonal or a monoclinic elastic symmetry is discussed. Four cell geometries, one isometric and three distorted, were manufactured using masked stereolithography 3D printing technology and measured in a laboratory compression set-up. Results from numerical simulations are compared to the experimental in terms of the compressive modulus.