High-resolution inelastic neutron scattering and extensive first-principles calculations have been used to explore the low-temperature phase of the hybrid solar-cell material methylammonium lead iodide up to the well-known phase transition to the tetragonal phase at ca. 160 K. Contrary to original expectation, we find that the Pnma structure for this phase can only provide a qualitative description of the geometry and underlying motions of the organic cation. A substantial lowering of the local symmetry inside the perovskite cage leads to an improved atomistic model that can account for all available spectroscopic and thermodynamic data, both at low temperatures and in the vicinity of the aforementioned phase transition. Further and detailed analysis of the first-principles calculations reveals that large-amplitude distortions of the inorganic framework are driven by both zero-point-energy fluctuations and thermally activated cation motions. These effects are significant down to liquid-helium temperatures. For this important class of technological materials, this work brings to the fore the pressing need to bridge the gap between the long-range order seen by crystallographic methods and the local environment around the organic cation probed by neutron spectroscopy.