The electron paramagnetic resonance (EPR) spectrum of the Mg acceptor is studied in a variety of GaN samples, including mm-thick free-standing substrates and sub-micron heteroepitaxial films. The former allows a view of the acceptor unique for EPR - in an environment with less than 107cm-3 dislocations and doping densities ranging from 4 × 1016 to 6 × 1018 cm-3. By probing Mg in a broad range of samples in one study a new feature of the acceptor emerges. The EPR data reveal an anisotropic line shape that reflects a change in the crystal field in the vicinity of the Mg acceptor. This feature must be accommodated by any of the models proposed for the Mg acceptor. Here, we show that one such previously proposed model agrees well with the EPR data obtained from the wide variety of samples studied. The work implies that Mg-doped GaN contains a common Mg-related defect which can be affected by local crystal fields established during growth.