Hybrid perovskites are a rapidly growing research area, having reached photovoltaic power conversion efficiencies of over 25 %. 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 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 all-inorganic analogue to the hybrid perovskite, FrPbBr$_3$, show that the large amplitudes of the scissoring modes are coupled to an opening of the electronic band gap. Finally, we use density functional theory calculations to show that the organic MA cations reorientate to accomodate the large amplitude scissoring modes.
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