  We consider the structure group of a non-degenerate symmetric (non-trivial)
set-theoretical solution of the quantum Yang-Baxter equation. This is a
Bieberbach group and also a Garside group. We show this group is not
bi-orderable, that is it does not admit a total order which is invariant under
left and right multiplication. Regarding the existence of a left invariant
total ordering, there is a great diversity. There exist structure groups with
space of left orders homeomorphic to the Cantor set and all left orders
Conradian, while there exist others that are even not unique product groups.
