  Let $G$ be a connected graph. The Jacobian group (also known as the Picard
group or sandpile group) of $G$ is a finite abelian group whose cardinality
equals the number of spanning trees of $G$. The Jacobian group admits a
canonical simply transitive action on the set $\mathcal{R}(G)$ of cycle-cocycle
reversal classes of orientations of $G$. Hence one can construct combinatorial
bijections between spanning trees of $G$ and $\mathcal{R}(G)$ to build
connections between spanning trees and the Jacobian group. The BBY bijections
and the Bernardi bijections are two important examples. In this paper, we
construct a new family of such bijections that includes both. Our bijections
depend on a pair of atlases (different from the ones in manifold theory) that
abstract and generalize certain common features of the two known bijections.
The definitions of these atlases are derived from triangulations and
dissections of the Lawrence polytopes associated to $G$. The acyclic cycle
signatures and cocycle signatures used to define the BBY bijections correspond
to regular triangulations. Our bijections can extend to subgraph-orientation
correspondences. Most of our results hold for regular matroids. We present our
work in the language of fourientations, which are a generalization of
orientations.
