Answer: The Pic⁰(G)-torsor structure is independent of the vertex v if and only if G is a planar ribbon graph.

This is the main theorem of "Rotor-routing and spanning trees on planar graphs", by Melody Chan, Thomas Church, and Joshua Grochow, which we just posted to the arXiv. Quoting from the introduction:

The proof is based on three key ideas. First, the rotor-routing action of the sandpile group on spanning trees can be partially modeled via rotor-routing on unicycles ([HLMPPW, §3]). This is a related dynamical system with the property that rotor-routing becomes periodic, rather than terminating after finitely many steps.

The second main idea is that the independence of the sandpile action on spanning trees can be described in terms of reversibility of cycles. We introduce the notion of reversibility (previously considered in [HLMPPW] only for planar graphs), and prove that reversibility is a well-defined property of cycles in a ribbon graph. We also establish a relation between reversibility and basepoint-independence.

Third, reversibility is closely related to whether a cycle separates the surface corresponding to the ribbon graph into two components. We prove that these conditions a��re almost equivalent. Moreover, although they are not equivalent for individual cycles, we prove that all cycles are reversible if and only if all cycles are separating, in which case the ribbon graph is planar.

We're grateful to Jordan for the question, which turned out to have a much more interesting answer than we expected! We're also grateful to Math Overflow for providing a venue for this question.

Answer: The Pic⁰(G)-torsor structure is independent of the vertex v if and only if G is a planar ribbon graph.

This is the main theorem of "Rotor-routing and spanning trees on planar graphs", by Melody Chan, Thomas Church, and Joshua Grochow, which we just posted to the arXiv [later published in IMRN]. Quoting from the introduction:

The proof is based on three key ideas. First, the rotor-routing action of the sandpile group on spanning trees can be partially modeled via rotor-routing on unicycles ([HLMPPW, §3]). This is a related dynamical system with the property that rotor-routing becomes periodic, rather than terminating after finitely many steps.

The second main idea is that the independence of the sandpile action on spanning trees can be described in terms of reversibility of cycles. We introduce the notion of reversibility (previously considered in [HLMPPW] only for planar graphs), and prove that reversibility is a well-defined property of cycles in a ribbon graph. We also establish a relation between reversibility and basepoint-independence.

Third, reversibility is closely related to whether a cycle separates the surface corresponding to the ribbon graph into two components. We prove that these conditions are almost equivalent. Moreover, although they are not equivalent for individual cycles, we prove that all cycles are reversible if and only if all cycles are separating, in which case the ribbon graph is planar.

We're grateful to Jordan for the question, which turned out to have a much more interesting answer than we expected! We're also grateful to Math Overflow for providing a venue for this question.