Ab initio calculations on (10,10) and (12,12) single-wall carbon-nanotube bundles show that the nature of the phase transformation under hydrostatic pressure is determined by the symmetry of the nanotubes. Bundles of (10,10) nanotubes that are incommensurate with the hexagonal lattice, have small deviations from hexagonal symmetry of the lattice even at zero pressure. A transition to monoclinic structure is obtained at about 1 GPa within the generalized gradient approximation such that the nanotubes transform to an oval shape. However, in the local-density approximation the monoclinic phase is retained even at zero pressure once the transformation has occurred. Bundles of (12,12) nanotubes are commensurate with the hexagonal symmetry of the lattice and show no transition even up to 6 GPa pressure except for a polygonization of the initially cylindrical nanotubes into a hexagonal shape. These results would resolve the contradictory conclusions obtained from experiments. {\textcopyright} 2002 The American Physical Society.