The gaseous flow of monoatomic Argon in a double-sided lid-driven square cavity is investigated using the direct simulation Monte Carlo method for different degrees of rarefaction. The effect of the direction of wall motion and the magnitude of wall velocities on the flow physics are analyzed. Unlike the single-sided cavity flow, the double-sided cavity flow generates different vortex formations especially for the parallel wall motion of the wall. The problem, therefore, merits a thorough study, which is attempted in the present paper using the direct simulation Monte Carlo method. Certain complex flow phenomena which are not captured using the numerical methods for continuum flows are revealed by the current method employed in the study. Two counter-rotating vortices are observed for the parallel wall motion whereas only one primary vortex can be observed for the antiparallel case. The variation in the flow and thermal properties is found to be significant at the onset of the transition regime and much smaller in the free molecular regime.