For the first time, experimental results have been obtained on a transition-metal tetroxy anion stabilized without any associated defects in a cubic crystal: the (ReO₄)^2- ion in KCI. Our EPR study, including experiments under uniaxial stress and static electric fields applied to the crystal, shows that a strong and essentially intramolecular Jahn-Teller effect occurs but not exclusively of the E⊗ε type. Appropriate linear combinations of the T2 vibrations involving a pseudo-Jahn-Teller effect and spin-orbit coupling must play an important part.