We present the results of a computational study on both WH6 and WF6 using a relativistic pseudopotential on tungsten, comparing results at the HF and MP2 levels with density functional calculations using local (SVWN), nonlocal (BLYP), and hybrid HF-DFT (ACM) functionals. In agreement with the earlier all-electron HF study of Schaefer and co-workers (J. Chem. Phys. 1993, 98, 508), we find that the ground state geometry of WH6 is far from octahedral; instead, there are four low-lying structures of C-3v (2) and C-5v (2) symmetry. Barrier heights for interconversion of these isomers are low, indicating a highly fluxional molecule. In contrast, the octahedral structure appears to be the only stable species on the ground state potential energy surface of WF6. It derives its stability from the greater ionic character of the W-F bond compared to that of W-H.