We investigate the nature of higher-order effects arising in solid-state nuclear magnetic resonance (NMR) when quadrupolar nuclei are subject to significant chemical shift anisotropies. It is shown that the quadrupole interaction can give rise to shielding-derived terms that are not entirely averaged away by conventional magic-angle spinning (MAS). These terms are proportional to the square of the z component of the spin angular momentum and therefore leave unaffected both the central and other -m(I)<---->+m(I) symmetric multiple-quantum transitions, yet lead to noticeable effects when monitoring other nonsymmetric transitions within the spin manifold. The recently-developed satellite-transition (ST) MAS NMR method for the simultaneous averaging of the first- and second-order quadrupole effects makes such quadrupole-shielding cross terms observable. Although this may present a resolution limitation to this averaging scheme, it opens up new possibilities for determining the coupling parameters of the quadrupolar nucleus-particularly the relative orientation between its quadrupole and shielding tensors. Average Hamiltonian derivations of these effects are explored, and employed to derive analytical expressions for their resulting splittings. These predictions are then successfully compared with variable-field STMAS NMR spectra of a Co-59-containing sample. A brief discussion of potential complications arising from third-order quadrupole effects when trying to analyze such line shapes is also presented. (C) 2003 American Institute of Physics.