In the asymptotic theory of non-linear differential equations it is frequently important to have a precise asymptotic description not only of the elements of the coefficient field, but also of the real parts of these elements. To secure such a description the present paper introduces into the theory of graduated fields a concept of Schwarzian symmetry, and a certain concept of topological closure, and, depending upon both of these, a concept of essential symmetry. The central result presented is the following theorem If F sub 0 is essentially symmetric, so is the algebraic closure of F sub 0. This is proved in an abstract setting which uses and develops the authors theory of graduated fields, as introduced in On the algebraic closure of certain partially ordered fields. Trans. Amer. Math. Soc. vol. 105 1962 pp. 229-250.