Ab initio all-electron relativistic Dirac-Fock (DF) and nonrelativistic (NR) Hartree-Fock (HF) self-consistent field (SCF) molecular calculations are reported for the tetrahedral superheavy hassium tetroxide (HsO(4)) and its lighter congener OsO4 at various M-O (M=Hs,Os) bond lengths. Our relativistic DF and NR HF wave functions predict the ground state of the tetrahedral HsO(4) and OsO4 to be bound and the predicted relativistic (nonrelativistic) atomization energy (A(e)) for HsO(4) and OsO4 is 15.35 (6.83) and 10.32 (5.59) eV, respectively. Relativistic effects lead to dramatic increase of similar to225% and 185% in the predicted A(e) of HsO(4) and OsO4, respectively. Mulliken population analysis of our relativistic DF (NR HF) wave functions for the HsO(4) and OsO4 yields a charge on Hs of +1.06 (+1.14) and Os of +0.88 (+1.94), respectively. Our DF SCF wave function, in contrast to our NR HF wave function, predicts HsO(4) to be more ionic, i.e., less volatile than OsO4 and our prediction is in agreement with the recent experimental work for the superheavy element hassium (http://enews.lbl.gov/Science-Articles/Archive/108-chemistry.html, dated 4 June 2001).