Internally 4-connected binary matroids with cyclically sequential orderings

We characterize all internally 4-connected binary matroids MM with the property that the ground set of MM can be ordered (e0,…,en−1)(e0,…,en−1) in such a way that {ei,…,ei+t}{ei,…,ei+t} is 4-separating for all 0≤i,t≤n−10≤i,t≤n−1 (all subscripts are read modulo nn). We prove that in this case either n≤7n≤7 or, up to duality, MM is isomorphic to the polygon matroid of a cubic or quartic planar ladder, the polygon matroid of a cubic or quartic Möbius ladder, a particular single-element extension of a wheel, or a particular single-element extension of the bond matroid of a cubic ladder.