The optical polarization anisotropy of quantum wells (QW) in structures fabricated from materials with cubic symmetry due to the anisotropic character of the Luttinger Hamiltonian for the valence band is studied. The matrix elements of interband optical transitions are shown to be different for different orientations of the linear polarization of light, oriented in the plane of a structure, provided that the structure growth axis does not coincide with any of the main crystallographic directions, i.e., with either [100],[010],[001] or [111]. The degree of this in-plane anisotropy of the polarization is calculated theoretically within an appropriate model for arbitrary structure growth orientation. The polarization anisotropy is experimentally detected for the fundamental 1hh-1e transition by studying the reflectivity from [120]-oriented Cd 0 . 8 Mn 0 . 2 Te/CdTe/Cd 0 . 8 Mn 0 . 2 Te QWs. At the same time the effect is shown to be absent in the case of QW grown along [100] direction. The measured polarization anisotropy in the former case shows a predicted π-periodicity as the polarization direction is rotated within the plane of the QW. The magnitude of the effect is also in qualitative agreement with the model. Quantitatively, the measured values tend to be some what larger than those predicted by the model. Surprisingly, the polarization anisotropy of the reflection by the barrier excitons is also detected. For the [120]-oriented structures this anisotropy has, approximately, the same magnitude as the one of the exciton transitions in the QWs. In the structure grown along [100], the anisotropy of the reflectivity of the barrier exciton transitions is also observed, although it is much smaller in magnitude. Some hypothetical explanations of these observations are put forward.