Theoretical calculations are performed to study transformations in silica as a function of nonhydrostatic stress. Molecular-dynamics calculations reveal a crystalline-to-crystalline transition from alpha-quartz to a phase with five-coordinated silicon (Si-V) at high pressure in the presence of deviatoric stress. The phase, which appears for specific orientations of the stress tensor relative to the crystallographic axes of quartz, is a crystalline polymorph of silica with five-coordinated silicon. The structure possesses P3(2)21 space-group symmetry. First-principles calculations within the local-density approximation, as well as molecular dynamics and energy minimization with interatomic potentials, find this phase to be mechanically and energetically stable with respect to quartz at high pressure. The calculated x-ray diffraction pattern and vibrational properties of the phase are reported. Upon decompression, the Si-V phase reverts to alpha-quartz through an intermediate four-coordinated phase and an unusual isosymmetrical phase transformation. The results suggest the importance of application of nonhydrostatic stress conditions in the design and synthesis of novel materials.