  Cyclic polytopes are characterized as simplicial polytopes satisfying Gale's
evenness condition (a combinatorial condition on facets relative to a fixed
ordering of the vertices). Periodically-cyclic polytopes are polytopes for
which certain subpolytopes are cyclic. Bisztriczky discovered a class of
periodically-cyclic polytopes that also satisfy Gale's evenness condition. The
faces of these polytopes are braxtopes, a certain class of nonsimplicial
polytopes studied by the authors. In this paper we prove that the
periodically-cyclic Gale polytopes of Bisztriczky are exactly the polytopes
that satisfy Gale's evenness condition and are braxial (all faces are
braxtopes). The existence of other periodically-cyclic Gale polytopes is open.
