The first example of a pressure-induced Jahn-Teller distortion switch has been observed in a single-crystal time-of-flight neutron diffraction study of the Tutton salt (ND4)2[CuD2O6](SO4)2 at T = 15 K, P = 1 bar (D(L) and T = 15 K, P = 1.5 kbar (D(H)). Also studied was the hydrogenous salt (NH4)2[Cu(H2O)6](SO4)2 at T = 14 K, P = 1.4 kbar (H(H)). The unit cell parameters are as follows: D(L), a = 9.451(2) angstrom, b = 12.736(3) angstrom, c = 6.096(1) angstrom, beta = 107.13(2)degrees, Z = 2, V = 701.2(3) angstrom3, space group = P2(1)/a; D(H), a = 9.136(i) angstrom, b = 12.285(2) angstrom, c = 6.371(1) angstrom, beta = 106.28(2)degrees, Z = 2, V = 686.4(2) angstrom3, space group = P2(1)/a; H(H), a = 9.068(2) angstrom, b = 12.232(2) angstrom, c = 6.340(1) angstrom, beta = 106.44(2), Z = 2, V = 674.5(2) angstrom3, space group = P2(1)/a. Application of 1.5 kbar of pressure to the deuterated crystal produces a decrease in the lengths of the a and b axes by 0.315(2) and 0.451(4) angstrom, respectively, whereas the c axis increases by 0.275(1) angstrom. Comparison of the D(L) and D(H) structures shows that the long axis of the Jahn-Teller distortion has switched by 90-degrees, i.e., Cu(1)-O(7) = 2.022(2) and Cu(1)-O(8) = 2.310(2) angstrom for D(L) vs Cu(1)-O(7) = 2.290(2) and Cu(1)-O(8) = 2.014(2) angstrom for D(H), so that the D(H) and H(H) structures are similar. For H(H) Cu(1)-O(7) = 2.272(2) and Cu(1)-O(8) = 2.005(2) angstrom. The switching of the long axis of the Jahn-Teller distortion appears to be associated with the rotation of the ND4+ ion with a concomitant change in the hydrogen bonding of the coordinated water molecules with the SO42- ions. The resulting adiabatic potential energy surfaces are calculated using tetragonal and orthorhombic strain parameters estimated from the temperature variation of the Cu-O bond lengths.