The topology of the ground-state potential energy surface of M(CN)(6) with orbitally degenerate T-2(2g) (M = Ti-III (t(2g)(1)), Fe-III and Mn-II (both low-spin t(2g)(5))) and T-3(1g) ground states (M = V-III (t(2g)(2)), Mn-III and Cr-Il (both low-spin t(2g)(4))) has been studied with linear and quadratic Jahn-Teller coupling models in the five-dimensional space of the E. and tau(2g) octahedral vibrations (T-g circle times(epsilon(g)+tau(2g) ) Jahn-Teller coupling problem (Tg = T-2(2g), T-3(1g))). A procedure is proposed to give access to all vibronic coupling parameters from geometry optimization with density functional theory (DFT) and the energies of a restricted number of Slater determinants, derived from electron replacements within the t(2g)(1,5) or t(2g)(2,4) ground-state electronic configurations. The results show that coupling to the tau(2g) bending mode is dominant and leads to a stabilization of D-3d structures (absolute minima on the ground-state potential energy surface) for all complexes considered, except for [Ti(CN)(6)](3-), where the minimum is of D-4h symmetry. The Jahn-Teller stabilization energies for the D3d minima are found to increase in the order of increasing CN-M pi back-donation (Ti-III < V-III < Mn-III < Fe-III < Mn-II < Cr-II). With the angular overlap model and bonding parameters derived from angular distortions, which correspond to the stable D3d minima, the effect of configuration interaction and spin-orbit coupling on the ground-state potential energy surface is explored. This approach is used to correlate Jahn-Teller distortion parameters with structures from X-ray diffraction data. Jahn-Teller coupling to trigonal modes is also used to reinterpret the anisotropy of magnetic susceptibilities and g tensors of [Fe(CN)6]3-, and the T-3(1g) ground-state splitting of [Mn(CN)(6)](3-), deduced from near-IR spectra. The implications of the pseudo Jahn-Teller coupling due to t(2g)-e(g) orbital mixing via the trigonal modes (tau(2g)) and the effect of the dynamic Jahn-Teller coupling on the magnetic susceptibilities and g tensors of [Fe(CN)(6)](3-) are also addressed.