Reversing double-step strains provide a severe test of constitutive equations and have often been used to test the Doi-Edwards (DE) model with and without the independent alignment approximation. We report measurements of the full stress tensor in a concentrated monodisperse polystyrene solution subjected to reversing double-step strains using flow birefringence. Shear stress and first normal stress difference results agree with previous studies. In flows where the second strain is half the magnitude of the first ("specialized type-B"), certain rheological models predict that normal stresses should be independent of time between strains (t(1)), and equal to those measured in single-step strain. Both first and second normal stress differences follow this behavior at long times, but deviations are found at short times, where chain retraction is occurring. In flows with equal and opposite step strains ("type-C"), the ratio N-1/sigma is found to be equal to the strain. In both of these flows, the DE model predicts the normal stress ratio -N-2/N-1 to be independent of t(1). In specialized type-B flows, the experimental normal stress ratio is nearly independent of t(1), while in type-C flows the ratio depends strongly on t(1) and is found to be substantially larger than that predicted by the DE model or observed in single-step strains. DE calculations for more general flow protocols predict that the normal stress ratio may depend on t(1) and on the time following the second deformation. Experiments show qualitative agreement with both forms of the DE model for these other flows, where the major influence of the independent alignment approximation is on the magnitude of the predicted normal stress ratio.