The multiferroic spinel is a benchmark material for exploring the competition of spin-orbit (SO) and Jahn-Teller (JT) coupling. Our magnetic and thermodynamic studies of stoichiometric single-crystalline samples evidence a magnetic-field-induced spin-reorientation transition in the cooperative JT state below 10 K. At 2 K, at a critical magnetic field of 4.5 T, the magnetization measured along the hard magnetization axis manifests a jump to the fully saturated state accompanied by a steplike decrease of the sound velocity and an abrupt increase of the magnetostriction. All these quantities reveal a hysteretic behavior pointing towards a first-order magnetostructural transformation. Below the JT transition, the specific heat shows a complex behavior upon the application of magnetic fields depending on the crystallographic directions. The observed reduction by 20% of the magnetic anisotropy below the JT transition is attributed to the competition of the SO and JT interactions tuned by external magnetic fields. The concomitant change of the structural symmetry results in a change of the splitting of the lowest levels of the doublet of the tetrahedrally coordinated ions.