Methylammonium lead iodide perovskite can make high-efficiency solar cells, which also show an unexplained photocurrent hysteresis dependent on the device-poling history. Here we report quasielastic neutron scattering measurements showing that dipolar CH3NH3+ ions reorientate between the faces, corners or edges of the pseudo-cubic lattice cages in CH3NH3PbI3 crystals with a room temperature residence time of ~14 ps. Free rotation, π-flips and ionic diffusion are ruled out within a 1–200-ps time window. Monte Carlo simulations of interacting CH3NH3+ dipoles realigning within a 3D lattice suggest that the scattering measurements may be explained by the stabilization of CH3NH3+ in either antiferroelectric or ferroelectric domains. Collective realignment of CH3NH3+ to screen a device’s built-in potential could reduce photovoltaic performance. However, we estimate the timescale for a domain wall to traverse a typical device to be ~0.1–1 ms, faster than most observed hysteresis.