  For a family of polytopes of even dimension $2p$, known as
\textit{dual-neighborly}, it has been shown for $p\ne 2$ that the associated
intersection of quadrics is a connected sum of sphere products $S^p\times S^p$.
In this article we give formulas for the number of terms in that connected sum.
Certain combinatorial operations produce new polytopes whose associated
intersections are also connected sums of sphere products and we give also
formulas for their number. These include a large amount of simple polytopes,
including many odd-dimensional ones.
