Attractive colloidal glasses are unique as their dynamical arrest is a combination of entropic crowding effects and energetic bonds formation. When such systems are subjected to flow, their dynamics are activated in a way which differs from hard-sphere glasses that melt through a "convective cage release mechanism." Here, we investigate the microscopic dynamics by measuring the relaxation spectrum during flow using orthogonal superposition rheometry. A small amplitude oscillatory strain is imposed perpendicularly to a steady shear flow, and superposition moduli are measured. Brownian dynamic simulations are utilized complementary to extract the moduli from mean-squared displacements using the generalized Stokes-Einstein relation. At low Peclet number, a crossover frequency between elastic and viscous moduli is detected, representing the relaxation time associated with shear-induced particles escape from their frustration (localization) under flow. For the repulsive glass, this is related to shear-induced cage renewal of particles. For attractive glasses, however, when particles escape their localized length (bonds), they move with no further hindrance with the escape time, which is independent of attraction strength and interestingly faster than that in the repulsive glass. This is attributed to particle localization at shorter length scales due to bonding. At high Peclet, a second low frequency crossover is observed and a low frequency elastic dominated response emerges. This elastic response may originate from slow relaxation of hydroclusters or be a consequence of more intricate nature of superposition moduli. At high frequencies, both orthogonal moduli increase relative to quiescent state due to shear-induced cage deformation, which slows down in-cage dynamics. (C) 2019 The Society of Rheology.