The high-pressure crystal structure evolution of CH3NH3PbBr3 (MAPbBr3) perovskite has been investigated by single-crystal X-ray diffraction and synchrotron-based powder X-ray diffraction. Single-crystal X-ray diffraction reveals that the crystal structure of MAPbBr3 undergoes two phase transitions following the space-group sequence: Pm3̅m → Im3̅ → Pmn21, unveiling the occurrence of a nonpolar/polar transition (Im3̅ → Pmn21). The transitions take place at around 0.8 and 1.8 GPa, respectively. This result contradicts the previously reported phase transition sequence: Pm3̅m → Im3̅ →Pnma. In this work, the crystal structures of each of the three phases are determined from single-crystal X-ray diffraction analysis, which is later supported by Rietveld refinement of powder X-ray diffraction patterns. The pressure dependence of the crystal lattice parameters and unit-cell volumes are determined from the two aforementioned techniques, as well as the bulk moduli for each phase. The bandgap behavior of MAPbBr3 has been studied up to around 4 GPa, by means of single-crystal optical absorption experiments. The evolution of the bandgap has been well explained using the pressure dependence of the Pb-Br bond distance and Pb-Br-Pb angles as determined from single-crystal X-ray diffraction experiments.