A braided link is the link formed by a braid closure along with the braid's axis. The bi-orderability of such links is equivalent to asking whether or not the free group automorphism induced by the braid preserves a bi-ordering of the free group. Kin-Rolfsen produced many examples of both order-preserving and non-order-preserving braids including a complete classification of periodic braids. In this talk, we'll survey some key results and techniques for finding order-preserving braids and obstructing order-preservingness. This includes work from an ongoing project with Nancy Scherich and Hannah Turner. This work is partially funded by the NSF grant DMS-2213213. Part of this research was conducted using computational resources and services at the Center for Computation and Visualization, Brown University.