  Results of Perron and Rolfsen imply that untwisted hyperbolic once-punctured
torus bundles over the circle have bi-orderable fundamental groups. They do
this by showing that the action of the monodromy preserves a "standard"
bi-ordering formed using the lower central series of the free group. Here we
investigate other bi-orderings that punctured torus bundle groups can have. We
show that for every such bi-ordering, the largest and second largest proper
convex subgroups match the corresponding convex subgroups for a standard
bi-ordering. Moreover, if there exists a third largest convex subgroup, it must
also match the third largest convex subgroup for a standard bi-ordering.
However, we also show that these groups admit non-standard bi-orderings.
