We investigate the yielding and flow of hard-sphere colloidal glasses by combining rheological measurements with the technique of light scattering echo. The poly-methylmethacrylate particles used are sufficiently polydisperse that crystallization is suppressed. Creep and recovery measurements show that the glasses can tolerate surprisingly large strains, up to at least 15%, before yielding irreversibly. We attribute this behaviour to 'cage elasticity', the ability of a particle and its cage of neighbours to retain their identity under quite large distortion. Results from light scattering echo, which measures the extent of irreversible particle rearrangement under oscillatory shear, support the notion of cage elasticity. In the lower concentration glasses we find that particle trajectories are partly reversible under strains which significantly exceed the yield strain.