The properties of piezoelectric metamaterials are numerically investigated using the examples of symmetric and asymmetric Gibson–Ashby cells with medium and high porosity. The paper demonstrates significant influence of the cell geometry and polarization model on the values of the effective properties and the resulting anisotropy type. Namely, a uniformly polarized symmetric cell made of a piezoceramic material of the hexagonal anisotropy class of 6mm has the effective moduli of the tetragonal symmetry class of 4mm, and a similar asymmetric cell has the effective moduli of the rhombic symmetry class of 2mm. A similar result is observed in piezoceramic Gibson–Ashby cells featuring piecewise uniform polarization models, where piezoceramic blocks are polarized in the air along one axis. However, simultaneous or sequential polarization of the cells along different directions can lead to the effective moduli matrices that differ from the standard syngony types. The paper also explores the existence of a complete or partial auxetic piezoelectric effect in such structures.