On Gale and braxial polytopes

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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.