EPR experiments have been carried out on irradiated KBr crystals doped with ReO₄^- ions at 4.2K. They indicate the formation of ReO₄^2- (5d¹, ²E ground state) subjected to an E⊗ε-type Jahn-Teller distortion. Uniaxial stress experiment show that the ReO₄^2- ion reorientates locally and the spin Hamiltonian parameters are unaltered by the application of the stress. A quadratic stress coupling term cannot be neglected. A model based on the approach of M.C.M. O'Brien is proposed to explain the experimental results.