The piezoelectricity law is a constitutive model that describes how mechanical and electric fields are coupled within a material. In its linear formulation this law comprises three constitutive tensors of increasing order: the second order permittivity tensor S, the third order piezoelectricity tensor P and the fourth-order elasticity tensor C. In a first part of the paper, the symmetry classes of the piezoelectricity tensor alone are investigated. Using a new approach based on the use of the so-called clips operations, we establish the 16 symmetry classes of this tensor and provide their associated normal forms. Second order orthogonal transformations (plane symmetries and π-angle rotations) are then used to characterize and classify directly 11 out of the 16 symmetry classes of the piezoelectricity tensor. An additional step to distinguish the remaining classes is proposed