The idea of writing supergravity as a double copy of super Yang-Mills theories has proved a fruitful one, most notably in the context of scattering amplitudes. I will explore this idea of "squaring" at a fundamental level to explain how the symmetries of the former(both local and global) can be written as a double copy of the latter. I will show how the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry are obtained from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincare. We give a gravity-gauge dictionary by convoluting fields and parameters. At the global level, I will explain how the coset groups of supergravity can be derived from the global R-symmetries of the SYM multiplets, via the 4 division algebras:reals, complexes, quaternions and octonions. Finally I will explore some intriguing possible applications of our dictonary.