The authors present a group-theoretical characterization of elastic polycrystalline materials. In order to achieve this, the constitutive equation for Cauchy stress is considered to depend not only on the deformation gradient but also on a set of so-called material “descriptors” (examples of which are lattice vectors of a unit cell of a crystallite, and the orientation distribution function – ODF – itself in a polycrystal). The group characterization is based on the assumption that, “in so far as the mechanical response is concerned, any two configurations related by a rigid displacement and with the same material descriptors are indistinguishable”. First, the authors propose a general and abstract setting. Then this is specialized to the case of polycrystalline materials in linear elasticity with initial stresses (original motivation of the work for acousto-elastic applications). Explicit computations are given for prestressed orthorhombic aggregates of cubic crystallites, which are further specialized to the case where the descriptors are composed of the initial stress tensor and of the ODF. This allows the authors to derive equations for fourth- and sixth-order (acoustoelastic) elasticity tensors. In particular, a model-independent formula is obtained for the latter. This formula clearly exhibits the effects of texture on elastic response. This very useful paper generalizes, complements, and somewhat supersedes a part of the results of a previous work by the second author [Arch. Ration. Mech. Anal. 143, No. 1, 77-103 (1998; Zbl 0917.73007)].