We report rheological and structural studies of a nematic surfactant solution (CPCl/hex) in transient shear flows. Upon step changes in steady shear flow conditions, the shear stress exhibits damped oscillations that scale with shear strain, attributed to director tumbling under shear. In situ small-angle x-ray scattering spectroscopy is used to measure changes in average micellar orientation during similar transient shear protocols; the measured orientation parameter also exhibits damped oscillations reflecting the underlying tumbling dynamics. Stress and structure data are compared with predictions of the Larson-Doi tumbling polydomain model. The model predictions are qualitatively similar to the experimental observations. In particular, orientation is observed to initially decrease upon shear flow reversal in CPCl/hex, in agreement with the Larson-Doi predictions but in contrast to previous results for model lyotropic solutions of poly(benzyl glutamate). Shear stress and average orientation are closely related during the oscillatory response to transient flows. The data suggest that higher orientation in the flow direction leads to low instantaneous stresses, and vice versa. The Larson-Doi model does not predict this relationship.