Abstract Hybrid perovskites are a rapidly growing research area, having reached photovoltaic power conversion efficiencies of over 25 %. There is a increasing consensus that the structures of these materials, and hence their electronic structures, can not be understood purely from the time and space averaged crystal structures observable by conventional methods. We apply a symmetry-motivated analysis method to analyse X-ray pair distribution function data of the cubic phases of the hybrid perovskites MAPb$X_3$ ($X$ = I, Br, Cl). We demonstrate that, even in the cubic phase, the local structure of the inorganic components of MAPb$X_3$ ($X$ = I, Br, Cl), are dominated by scissoring type deformations of the Pb$X_6$ octahedra. We find these modes to have a larger amplitude than equivalent distortions in the $A$-site deficient perovskite ScF$_3$ and demonstrate that they show a significant departure from the harmonic approximation. Calculations performed on an inorganic perovskite analogue, FrPbBr₃ show that the large amplitudes of the scissoring modes are coupled to a dynamic opening of the electronic band gap. Finally, we use density functional theory calculations to show that the organic MA cations reorientate to accommodate the large amplitude scissoring modes.