Mechanisms of thermoluminescence (TL) in CaSO₄:Dy are studied using electron spin resonance (ESR) in single-crystal samples. The ESR analysis indicates the presence of several variations of a distorted SO₄^- centre locally stabilised by a nearest neighbour Ca vacancy. This centre's structure, which has not been reported previously, relates to several centres observed in this system and could also explain results in related compounds. The presence of the Dy impurity, which increases the concentration of Ca vacancies due to charge neutrality requirements, enhances both the concentration of these centres and the intensity of the major TL peaks near 220 and 350 degrees C. The specific role of these and other previously reported centres in the TL of this material is followed using heat treatments. This enables one to identify centres that change in the range of the observed TL peaks as well as centres that seem to be related to one another by charge transfer or rearrangement. The behaviour of the observed centres and their relationship to the TL behaviour of the system reemphasises the complexity of this system.