Lead halide perovskites such as methylammonium lead triiodide (CH₃NH₃PbI₃) have outstanding optical and electronic properties for photovoltaic applications, yet a full understanding of how this solution-processable material works so well is currently missing. Previous research has revealed that CH₃NH₃PbI₃ possesses multiple forms of static disorder regardless of preparation method, which is surprising in light of its excellent performance. Using high energy resolution inelastic X-ray (HERIX) scattering, we measure phonon dispersions in CH₃NH₃PbI₃ and find direct evidence for another form of disorder in single crystals: large-amplitude anharmonic zone edge rotational instabilities of the PbI₆ octahedra that persist to room temperature and above, left over from structural phase transitions that take place tens to hundreds of degrees below. Phonon calculations show that the orientations of the methylammonium (CH₃NH₃⁺) couple strongly and cooperatively to these modes. The result is a noncentrosymmetric, instantaneous local structure, which we observe in atomic pair distribution function (PDF) measurements. This local symmetry breaking is unobservable by Bragg diffraction but can explain key material properties such as the structural phase sequence, ultralow thermal transport, and large minority charge carrier lifetimes despite moderate carrier mobility. From the PDF we estimate the size of the fluctuating symmetry broken domains to be between 1 and 3 nm in diameter.