Abstract: We investigate the flow inside a two dimensional square cavity driven by the motion of two mutually facing walls independently sliding at different speeds. The exploration, which employs the lattice Boltzmann Method (LBM), extends on previous studies (An et al. 2019, 2020a, 2020b) that had the two lids moving with the exact same speed in opposite directions. Unlike, there, here the flow is governed by two Reynolds numbersassociated to the velocities of the two moving walls. For convenience, we define a bulk Reynolds number () and quantify the driving velocity asymmetry by a parameter. The parameterhas been defined in the range and a systematic sweep in Reynolds numbers has been undertaken to unfold the transitional dynamics path of the two-sided wall-driven cavity flow. In particular, the critical Reynolds numbers for Hopf and Neimark-Sacker bifurcations have been determined as a function of . The eventual advent of chaotic dynamics and the symmetry properties of the intervening solutions are also analyzed and discussed. The paper unfolds for the first time the full bifurcation scenario as a function of the two Reynolds numbers, and reveals the different flow topologies found along the transitional path.