Inelastic neutron scattering (INS) spectra are presented for chromium(II) Tutton salts, as a function of the temperature and pressure. Transitions are observed between the levels of the (5)A(g) (C-i) ground term and the data modeled with a conventional S = 2 spin Hamiltonian. At 10 K and ambient pressure, the zero-field-splitting parameters of the ammonium salt, (ND4)(2)Cr(D2O)(6)(SO4)(2), are determined as D = - 2.431( 4) cm(-1) and E = 0.091( 4) cm(-1), evolving to D = - 2.517( 4) cm(-1) and E = 0.127(5) cm(-1) upon application of 7.5(1.0) kbar of quasi-hydrostatic pressure. By contrast, the change in the INS spectrum of the rubidium salt in this pressure range is comparitively minor. The results are interpreted using a (5)Exe vibronic-coupling Hamiltonian, in which low-symmetry strain, perturbing the adiabatic potential-energy surface, is pressure-dependent. It is argued that, for the ammonium salt, the change with pressure of the anisotropic strain impinging upon the [Cr(D2O)(6)](2+) cation is sufficient to cause a switch of the long and intermediate Cr - OD2 bonds, with respect to the crystallographic axes.