  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.
