We present the results of a computational study of the low-energy dynamics of silica glass. Molecular dynamics simulation results show that parts of the glass structure can undergo large cooperative reorientations of SiO₄ tetrahedra. These motions involve reorientations of about 30 tetrahedra with an energy barrier of about 0.06 eV. We relate these motions to the presence of double-well potentials which give rise to two-level tunnelling states in the model, thereby providing the mechanism for the anomalous low-temperature thermal properties of glasses. Simulation of larger structures of silica glass shows that jump events become more frequent and uncorrelated with each other. In addition to studying the flexibility of silica glass in terms of the large tetrahedral rearrangements, we also address the flexibility of silica glass in terms of its ability to sustain low-ω floppy modes. The latter part of the study is supported by inelastic neutron scattering data, and we compare experimental and calculated dynamic structure factors in the energy range 0-10 meV and scattering vector range 0-8 Å^-1. By applying the analysis of the rigid-unit-mode model as initially developed for crystalline silicates to structures of silica glass we find that silica glass is surprisingly similar to its corresponding crystalline phases in its ability to support low-ω floppy modes. The same conclusion follows from the comparison of calculated vibrational densities of states of silica glass and its crystalline phases, and is borne out in the inelastic neutron scattering data.