Abstract A new set of invariants and symmetric and skew-symmetric second order tensor generators is introduced to represent the general form of a transversely isotropic constitutive law. In contrast to the standard system usually used in literature the new invariants and tensor generators have a well defined physical meaning which is strongly related to the deformation mechanisms of transversely isotropic materials. This substantially facilitates the decision which tensor generators should be considered in a constitutive law and on which invariants the scalar-valued functions should depend. Reducing the tensor function to a linear relation one gets a new frame independent representation of a fourth order transversely isotropic tensor. The relations between the different representations found in literature and that one introduced here are given. Their advantages are illustrated by the examples of elastic and inelastic constitutive laws of monocrystalline ice.