A continuum tensorial theory was formulated to describe the isothermal, incompressible flow of uniaxial rodlike nematic liquid crystalline polymers. The tensor theory was reduced to a vector theory that describes the microstructure of the uniaxial phase by specifying the director orientation and the scalar order parameter. The reduction establishes useful relationships between this theory and the Leslie and Ericksen theory. The model was solved for a given steady simple shear flow, assuming spatial homogeneity. Two types of orienting modes are predicted: (a) the simple aligning mode, in which the microstructure reaches a stable shear dependent steady state for all shear rates, and (b) the complex mode, which at sufficiently high nematic potentials exhibits three flow regimes (tumbling, oscillating, and stationary) according to the strength of the imposed shear flow. For sufficiently low nematic potentials, the complex mode predicts the existence of a single stationary mode. Bifurcation theory was used to fully characterize the existence and transitions between the three regimes of the complex mode. Guidelines for future simplifications of the constitutive equations are given. The frequently reported changes in the sign of normal stress differences with shear rate are captured by the model.