Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a total ordering (not to be confused with a well-ordering) which is "bi-translation invariant": a < b should imply cad < cbd.

Does anyone know any examples?

Totally ordered abelian groups are easy to come up with: any direct product of subgroups of the reals, with the lexicographic ordering, will do. Knowing some non-abelian ones would help reveal what aspects of totally ordered abelian groups really depend on them being abelian...