What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?

• In even dimensions, all facets of the dual are combinatorially equivalent. Are these facets themselves duals of cyclic polytopes?

• I think this cannot be true in odd dimensions $\ge 5$. For example, a 5-dimensional cyclic polytope is 2-neighborly, so its vertex figures are only 1-neighborly (is this true?), but 4-dimensional cyclic polytopes are 2-neighborly as well. So when does it happen that the facets are again duals of cyclic polytopes, and when are they all combinatorially equivalent?

• In general, is there some kind of classification of the combinatorial types of these facets?