A numerical investigation of vortex-induced rotations (VIRs) of a rigid square cylinder, which is free to rotate in the azimuthal direction in a two-dimensional uniform cross-flow, is presented. Two-dimensional simulations are performed in a range of Reynolds numbers between 45 and 150 with a fixed mass and moment of inertia of the cylinder. The parametric investigation reveals six different dynamic responses of the square cylinder (expanding on those reported by Zaki et al. (J. Fluids Struct., vol. 8, 1994, pp. 555–582)) and their coupled vortex patterns at low Reynolds numbers. In each characteristic regime, moment generating mechanisms are elucidated with investigations of instantaneous flow fields and surface pressure distributions at chosen time instants in a period of rotation response. Our simulation results also elucidate that VIRs significantly influence the statistics of drag and lift force coefficients: (i) the onset of a rapid increases of the two coefficients at $Re=80$ and (ii) their step increases in the autorotation regime.