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A theoretical study is undertaken into the Kerr nonlinear refraction observable in the transparency window in semiconductor heterostructures. Symmetry considerations indicate that the number of nonzero, independent tensor components increases from three to eight for single wavelength third-order nonlinear processes with the usual growth orientation. These are related to the observable phenomena of self-phase-modulation, cross-phase-modulation, and four-wave-mixing of the orthogonal polarization components in the usual waveguide geometry. For a 14:14 GaAs-AlAs superlattice, all eight independent elements are calculated. It is shown that the nonlinear response is highly anisotropic. The modification of the nonlinear coefficients with quantum well intermixing is investigated, with a several-fold reduction in value predicted. This modification could be exploited in some long-standing and novel nonlinear optical experiments.