  An exceptional point in the moduli space of compact Riemann surfaces is a
unique surface class whose full automorphism group acts with a triangular
signature. A surface admitting a conformal involution with quotient an elliptic
curve is called elliptic-hyperelliptic; one admitting an anticonformal
involution is called symmetric. In this paper, we determine, up to topological
conjugacy, the full group of conformal and anticonformal automorphisms of a
symmetric exceptional point in the elliptic-hyperelliptic locus. We determine
the number of ovals of any symmetry of such a surface. We show that while the
elliptic-hyperelliptic locus can contain an arbitrarily large number of
exceptional points, no more than four are symmetric.
