Christopher: what you actually described were the two main ways to take an arbitrary binary relation and turn it into a symmetric binary relation. These are the left and right adjoints to the forgetful functor from "sets with symmetric relation" to "sets with arbitrary binary relation". As usual the left adjoint is the "permissive, generous" method, which uses "or", while the right adjoint is the "cautious, ungenerous" method which uses "and".

You see, ["or" is itself a left adjoint, while "and" is a right adjoint](