  The Jacobian group ${\rm Jac}(G)$ of a finite graph $G$ is a group whose
cardinality is the number of spanning trees of $G$. $G$ also has a tropical
Jacobian which has the structure of a real torus; using the notion of break
divisors, An et al. obtained a polyhedral decomposition of the tropical
Jacobian where vertices and cells correspond to elements of ${\rm Jac}(G)$ and
spanning trees of $G$, respectively. We give a combinatorial description of
bijections coming from this geometric setting. This provides a new geometric
method for constructing bijections in combinatorics. We introduce a special
class of geometric bijections that we call edge ordering maps, which have good
algorithmic properties. Finally, we study the connection between our geometric
bijections and the class of bijections introduced by Bernardi; in particular we
prove a conjecture of Baker that planar Bernardi bijections are "geometric". We
also give sharpened versions of results by Baker and Wang on Bernardi torsors.
