A groupoid 21 = ζA; *> is called a left (resp. right) difference group if there is a binary operation + in A such that the system is an abelian group and x*y — —x + y (resp. x * y = x ~ y). A symmetric difference group is a groupoid satisfying all the identities common to both left and right difference groups. In this note we determine the structure of a symmetric difference group. Using this, we show that any finitely based equational theory of symmetric difference groups is one-based. This includes the known result that the theories of left and right difference groups are onebased. Other known results on finitely based theories of rings also follow.