The preference of XeF6 for either a trigonally distorted or a regular octahedral structure is determined by a delicate balance of several competing factors. A regular octahedron is favored (a) by electron correlation and (b) by the relativistic contraction of the Xe 5s orbital. In contrast, higher angular momentum (in particular f-type) basis functions on Xe favor a distortion. While earlier SCF or other nonrelativistic calculations were in apparent agreement with experimental evidence for a distorted structure, this has been due to a partial cancellation of errors. The present study contains all-electron calculations as well as calculations of the valence-electrons in an effective core potential. For the former, electron correlation has been included at the MP2 level and relativistic effects by means of direct perturbation theory, for the latter the highest level was CCSD(T) for the treatment of electron correlation, and relativistic effects were simulated by means of a quasirelativistic effective core potential. Both sets of calculations lead to consistent results. These indicate that the "XeF6-like" XF(6) compounds with light central atoms or ions like ClF6- or BrF6- prefer the structure of a regular octahedron. The same is true for KrF6, which is not stable with respect to Kr + 3F(2) but probably represents a local minimum. For these light central atoms, electron correlation is decisive for a regular structure, while at Hartree-Fock level, i.e., ignoring correlation effects, the structure of lowest energy is distorted. Regular octahedra are also predicted for systems with very heavy central atoms like RnF6, AtF6-, and PoF62-. For these compounds relativistic effects (supported by electron correlation) stabilize the regular octahedron. The situation is more complicated for XeF6, IF6-, and TeF62-. Here the strong distortion found at nonrelativistic SCF level is compensated partly, but apparently not completely, by electron correlation and relativistic effects. This results in distorted, fluctuating structures with only little stabilization compared to the regular octahedron. A similar situation holds for SeF62-. None of the available simple models of main-group structural chemistry is able to predict or rationalize all of these structures. One inevitably has to use more subtle descriptions.