We introduce PSN polytopes, whose k-skeleton is combinatorially equivalent to
that of a product of r simplices. They simultaneously generalize both
neighborly and neighborly cubical polytopes.

We construct PSN polytopes by three different methods, the most versatile of
which is an extension of Sanyal and Ziegler's "projecting deformed products"
construction to products of arbitrary simple polytopes. For general r and k,
the lowest dimension we achieve is 2k+r+1.