We report on the rheology of isotropic and nematic aqueous solutions of a sulfonated all-aromatic polyamide, poly(2,2 '-disulfonyl-4,4 '-benzidine terephthalamide) (PBDT), that forms high-aspect-ratio rod-like assemblies. For quiescently isotropic solutions, the concentration dependence of the zero-shear viscosity, longest relaxation time, and terminal modulus shows deviations in comparison to the Doi-Edwards theory for hard rods. For quiescently nematic solutions, we characterize the flow behavior through steady-state and transient nonlinear rheological measurements in conjunction with small-angle neutron scattering under shear. The steady-state flow curve is characterized by two anomalous shear thickening responses, one at moderate shear rates and the other immediately prior to flow alignment at high shear rates. We assign the origin of these shear thickening response to director "kayaking" and "out-of-plane steady" states, using predictions from prior high-resolution numerical simulations of sheared nematic rods. Utilizing transient shear flow reversals and step-down experiments, we characterize the oscillatory response of the nematic director through these flow regimes. When the first normal stress difference is plotted versus the shear stress during a transient step-down, the so-called dynamic stress path, the counterclockwise versus clockwise rotation has previously been shown to reveal the relative dominance of viscous versus elastic contributions to the stress tensor, respectively. Our measurements strongly suggest that the anomalous shear thickening behavior in nematic PBDT solutions arises from viscous stresses developed as the ensemble of rods undergoes periodic oscillatory motion under shear, rather than elastic stresses due to broadening of the orientational distribution function.