Bassanite (CaSO4.1/2 H2O) and anhydrite (CaSO4) are the low-temperature products of gypsum (CaSO4.2H2O) dehydration, which are obtained at about 373 and 433 K, respectively. These sulphates have non-centrosymmetric crystallographic point groups, but dielectric measurements do not reveal any piezo- or ferroelectric characteristic, and they practically behave like linear dielectrics. Positron lifetime spectra exhibit the existence of two different positron states, besides a free positron state. There is positronium formation in the three sulphates, and there is also evidence for the presence of a highly populated positron bound state which may be a complex state associated with positrons bound to SO42- ions. Parameter S estimated from the Doppler curve and the average positron lifetime show unquestionably the sensitivity of positrons to the phase transitions gypsum-bassanite and bassanite-anhydrite.