\textit{H. S. M. Coxeter} [Proc. Lond. Math. Soc., II. Ser. 43, 33-62 (1937; Zbl 0016.27101)] discovered that to every self-dual convex regular 4- polytope P, there may be associated a regular skew-polyhedron M. The automorphism group of M is that of P twisted by \(Z_ 2\). The authors define an incidence-polytope as a combinatorial analog of a polytope, and generalize Coxeter's construction to self-dual regular 4-incidence polytopes of type \(\{\) p,q,p\(\}\). Here ``type'' lists the orders of certain generators of the automorphism group of the polytope. The method uses twisting operations on certain groups. The associated regular map M is called the Petrie-Coxeter polyhedron of P. (Quotients of) Euclidean and hyperbolic tesselations may arise in this fashion.