Tensoring the field content of two super-Yang-Mills theories results in the field content of a certain supergravity theory, a procedure we call squaring. This thesis investigates how both the local and global internal symmetries enjoyed by the supergravity theory are inherited from the super-Yang-Mills factors. This is part of a much larger framework for studying a supergravity theory through its factorisation into simpler theories. The thesis begins by introducing local and global symmetries in the general context of relativistic field theory. The introduction is a short review on spacetime and internal symmetries in both gravitational and non-gravitational theories with particular focus on the supersymmetric regime. This is followed by the squaring idea and its appearance in various different contexts. After providing a unified description of all super-Yang-Mills theories over the four Normed Division Algebras R,C,H,O, the global internal supergravity symmetries are built with the help of the mathematical construction of the magic square, generalised to what we call the magic pyramid of supergravities. A physical interpretation of the formula reveals the Yang-Mills origin of the symmetries and demonstrates how simultaneous supersymmetry transformations on both factors can contribute to bosonic generators. The analysis is then extended to accommodate more exotic squarings by allowing for the coupling of matter multiplets to the super-Yang-Mills factors. Finally, the focus shifts to local internal symmetries whose linear form is derived directly from the corresponding linear factors. After a general treatment of off-shell squaring in various spacetime dimensions, the possibility of extending the construction to non-linear gravity is discussed.