A cyclic $d$-polytope is "neighborly," where every set of vertices sized $d/2$ together make up a face. What I'm interested in: in a cyclic $d$-polytope with $N$ facet ($d-1$ dimensional faces), does every facet "border" every other facet (i.e., share a $d-2$ dimensional face)?
Why I think it might: we can use the upper bound theorem in dual form (https://www.cs.mcgill.ca/~fukuda/soft/polyfaq/node12.html) to find the total number of $d-2$ dimensional faces for a $d$-polytope with $N$ facets. If you use the formula in the provided link, we find that there are $N(N-1)/2$ such $d-2$ dimensional faces. Since every such face borders $2$ facets, this total number of $d-2$ dimensional faces suggests that the property holds.