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Recently, a unique class of local Higher Spin Gravities with propagating massless fields in 4 d – Chiral Higher Spin Gravity – was given a covariant formulation both in flat and (A) d S 4 spacetimes at the level of equations of motion. We unfold the corresponding homological perturbation theory as to explicitly obtain all interaction vertices. The vertices reveal a remarkable simplicity after an appropriate change of variables. Similarly to formality theorems the A ∞ / L ∞ multi-linear products can be represented as integrals over a configuration space, which in our case is the space of convex polygons. The A ∞ -algebra underlying Chiral Theory is of pre-Calabi–Yau type. As a consequence, the equations of motion have the Poisson sigma-model form. [ABSTRACT FROM AUTHOR]