Single crystals of dimethyl sulfoxide (DMSO, (CH3)(2)SO) were in situ frozen at isochoric conditions in a diamond-anvil cell and their structures determined at 0.37, 0.56, and 2.4 GPa. At ambient pressure, DMSO freezes at 291 K in monoclinic phase alpha, space group P2(1)/c, stable in all the temperature range from its melting point down to 2 K. On increasing the pressure, DMSO freezes at 140 MPa/296 K in phase a too, but above 540 MPa it collapses into a more compact triclinic phase beta, space group P (1) over bar. The molecular aggregation in the crystal structure of DMSO is dominated by electrostatic attraction between negative and positive sites on the molecular surface and CH center dot center dot center dot O hydrogen bonds linking the molecules into dimers and chains. Most of this electrostatic matching and CH center dot center dot center dot O bonds present in phase alpha are preserved above 540 MPa, but the more tight packing in phase beta is achieved at the cost of broken dipole-dipole interactions between antiparallel S=O groups and repulsing contacts between electropositive H-atoms squeezed to distances commensurate with the sum of van der Waals radii. The isostructural relation between phases alpha and beta is strictly connected with the molecular aggregation governed by electrostatic matching of interatomic contacts.