Computer simulation of the core structure and glide of ordinary 1/2<110] dislocations and <101] superdislocations in L10 TiAl has been performed using the recently constructed BondOrder Potentials. This description of atomic interactions includes explicitly, within the tight-binding approximation, the most important aspects of the directional bonding, namely d-d, p-p and d-p bonds. The ordinary dislocation in the screw orientation was found to have a non-planar core and, therefore, high Peierls stress. The superdislocation was found to possess in the screw orientation either a planar (glissile) or a non-planar (sessile) core structure. However, the glissile core transforms into the sessile one for certain orientations of the applied stress. This implies a strong asymmetry of the yield stress and the break down of the Schmid law when the plastic flow is mediated by superdislocations. At the same time, this may explain the orientation dependence of the dislocation substructure observed in the single-phase γ-TiAl by electron microscopy.