Stereoisomers of C 13 Cl 2 exhibiting helical orbitals around a ring of carbon atoms were synthesized by atom manipulation on NaCl surfaces. We resolved the enantiomeric geometries of the singlet states by atomic force microscopy and mapped their helical orbital densities by scanning tunnelling microscopy. A π-orbital basis of the helical, non-planar singlets that twists by 90° in one circulation is consistent with a half-Möbius topology. In such a topology, the π-orbital basis changes sign with respect to two circumnavigations and is periodic with respect to four circumnavigations. A quasiparticle on a ring with this boundary condition could be interpreted as carrying a Berry phase of π/2. We demonstrate reversible switching of the topology, between the two singlets of oppositely threaded half-Möbius topology, and the planar, topologically trivial, triplet state. Multireference calculations, including large-scale sample-based ab initio calculations executed on quantum hardware, reveal that the switching is associated with a helical pseudo Jahn-Teller effect.