This paper sets down in detail the observation, first announced as a note (Romo-Uribe, A.; Windle, A. H. Macromolecules 1993, 26, 7100), of a strident and reversible orientation transition in a specially synthesized series of thermotropic random copolyester melts of the Vectra A900 type. The transition is between the flow-aligning regime normally seen in nematics and alignment of the global director at right angles to flow-alignment and parallel to the vorticity axis, which we refer to as the "log-rolling" regime. The orientation transition is observed in a comparatively narrow band of temperature above the crystal melting point, the width of this band decreasing with increasing strain rate and increasing molecular weight. We believe that observation of log-rolling in the main-chain thermotropic random copolymers provides confirmatory evidence for the occurrence of a smectic phase in isolated regions which are the precursors of the non periodic layer (npl) crystals which form on further cooling. The fact that the fog-rolling regime is not observed at high molecular weights and higher strain rates suggests that the low density of entanglements, which will occur in liquid crystalline polymers with less than completely rigid chains, will nevertheless compromise the log-rolling regime. Slow strain rates and lower molecular weights would each tend to enhance the relaxation of entanglements during shear and thus reduce the deleterious effect which entanglements would have on the log-rolling regime. Recent rheological measurements on the same set of polymers add support to this hypothesis.