Angle-resolved polarized transmission spectra of thin-film opals are studied experimentally and theoretically as a function of the azimuthal rotation of the plane of incidence at different angles of light incidence. In such 3-dimensional lattices, the refraction acquires the form of diffraction orders, each with a distinct spectrum, that propagate simultaneously along different directions. The corresponding continuous and patchy stop-bands in the photonic energy band structure of the opal photonic crystal are determined numerically. For diffraction at high Miller index planes, the diffraction pattern is often distorted due to multiple-wave diffraction. The finite stacking of hexagonally close-packed layers leads to an intrinsic 3-fold rotation symmetry of these spectra, which is particularly pronounced in transmission for frequencies that correspond to wave vectors within the 1st Brillouin zone. Despite the cubic symmetry of the opal lattice, cross-polarization coupling effects occur in the volume of the opal crystal. These effects are associated with the fact that, the opal eigenmodes are Bloch waves whose electromagnetic field distributions and dispersion relations are markedly different from plane waves.