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][1] (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...

  [1]: http://en.wikipedia.org/wiki/Total_order