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Let $M$ be a regular matroid. The Jacobian group ${\rm Jac}(M)$ of $M$ is afinite abelian group whose cardinality is equal to the number of bases of $M$.This group generalizes the definition of the Jacobian group (also known as thecritical group or sandpile group) ${\rm Jac}(G)$ of a graph $G$ (in which casebases of the corresponding regular matroid are spanning trees of $G$).  There are many explicit combinatorial bijections in the literature betweenthe Jacobian group of a graph ${\rm Jac}(G)$ and spanning trees. However, mostof the known bijections use vertices of $G$ in some essential way and areinherently "non-matroidal". In this paper, we construct a family of explicitand easy-to-describe bijections between the Jacobian group of a regular matroid$M$ and bases of $M$, many instances of which are new even in the case ofgraphs. We first describe our family of bijections in a purely combinatorialway in terms of orientations; more specifically, we prove that the Jacobiangroup of $M$ admits a canonical simply transitive action on the set ${\mathcalG}(M)$ of circuit-cocircuit reversal classes of $M$, and then define a familyof combinatorial bijections $\beta_{\sigma,\sigma^*}$ between ${\mathcal G}(M)$and bases of $M$. (Here $\sigma$ (resp. $\sigma^*$) is an acyclic signature ofthe set of circuits (resp. cocircuits) of $M$.) We then give a geometricinterpretation of each such map $\beta=\beta_{\sigma,\sigma^*}$ in terms ofzonotopal subdivisions which is used to verify that $\beta$ is indeed abijection.  Finally, we give a combinatorial interpretation of lattice points in thezonotope $Z$; by passing to dilations we obtain a new derivation of Stanley'sformula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$.

Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory