<p>A general approach is presented for inverting fourth-order material tensors expressed as a sum of products of rank-two tensors such as the identity tensor and structural tensors. With reference to transversely isotropic and orthotropic materials, the procedure exploits identities involving different tensor products of the structural tensors to obtain three expressions of the elasticity tensor in order to look for the one that can simplify as much as possible the fully intrinsic evaluation of the compliance tensor. The sets of elastic constants referred to the intrinsic expression of the elasticity and compliance tensor are mutually related and are expressed in an arbitrary reference frame, not necessarily aligned with the planes of material symmetry. Finally, to foster the use of the tensorial approach in numerical applications, we also derive explicit expressions relating elastic or compliance moduli both with the entries of the material stiffness matrix, since they are determined experimentally, and with engineering constants, since they are more often used in practice due to their direct mechanical meaning.</p>