We report studies on the average molecular orientation state in steady and transient shear flow of two lyotropic liquid crystalline polymers: poly(benzyl glutamate) [PBG] and hydroxypropylcellulose [HPC], both in m-cresol solution. An annular cone and plate X-ray shear cell is used to probe molecular orientation in the "1-2" plane, allowing simultaneous measurements of the degree of anisotropy and the average orientation angle relative to the flow direction. In steady shear flow, molecular orientation increases with shear rate in both materials. Comparisons with separate measurements in the "1-3" plane indicate that both materials exhibit a macroscopically biaxial orientation distribution function. The orientation angle is always small and exhibits a sign change from positive to negative values with increasing shear rate. In transient flows, anisotropy and orientation angle both exhibit damped oscillations that scale with shear strain. The Larson-Doi polydomain model is in qualitative agreement with data collected on PBG in flow reversal, but only after an initial transient response seen in the experiments over the first several strain units following the reversal. Following step-increase and step-decrease of shear rate, the Larson-Doi model makes qualitatively correct predictions of anisotropy but qualitatively incorrect predictions of the transient average orientation angle.