The structural and vibrational properties of SrWO4 have been studied experimentally up to 27 and 46 GPa, respectively, by angle-dispersive synchrotron X-ray diffraction and Raman spectroscopy measurements as well as using ab initio calculations. The existence of four polymorphs upon quasi-hydrostatic compression is reported. The three phase transitions were found at 11.5, 19.0, and 39.5 GPa. The ambient-pressure SrWO4 tetragonal scheelite-type structure (S.G. I41/a) undergoes a transition to a monoclinic fergusonite-type structure (S.G. I2/a) at 11.5 GPa with a 1.5% volume decrease. Subsequently, at 19.0 GPa, another structural transformation takes place. Our calculations indicate two possible post-fergusonite phases, one monoclinic and the other orthorhombic. In the diffraction experiments, we observed the theoretically predicted monoclinic LaTaO4-type phase coexisting with the fergusonite-type phase up to 27 GPa. The coexistence of the two phases and the large volume collapse at the transition confirm a kinetic hindrance typical of first-order phase transitions. Significant changes in Raman spectra suggest a third pressure-induced transition at 39.5 GPa. The conclusions extracted from the experiments are complemented and supported by ab initio calculations. Our data provides insight into the structural mechanism of the first transition, with the formation of two additional W-O contacts. The fergusonite-type phase can be therefore considered as a structural bridge between the scheelite structure, composed of [WO4] tetrahedra, and the new higher pressure phases, which contain [WO6] octahedra. All the observed phases are compatible with the high-pressure structural systematics predicted for ABO4 compounds using crystal-chemistry arguments such as the diagram proposed by Bastide.