"Snake" reconfiguration problem
Moving a snake from one position to some other is PSPACE complete.
Snake is trivially in PSPACE.
We give a PSPACE hardness reduction from Hearn's Nondeterministic Constraint Logic.

Nondeterministic Constraint Logic

Let a constraint graph be a directed graph with edges of weight $1$ and $2$, such that every vertex has incoming weight $\geq 2$.
Given two constraint graphs, it is PSPACE hard to transform one into the other by reversing edges, one at a time, keeping the incoming weight of each vertex $\geq 2$.
This still holds if the graph is planar and each vertex has degree $3$, and either $1$ or all $3$ of its edges have weight $2$.
The former type of vertex is called And and the latter is called Or (see figure).


Snake

In our construction, the snake's head will be chasing its tail at some small distance and will be forced to repeat the same cycle with minor modifications.
We embed each edge of the constraint graph as in the figure (edges shown in red), where we indicate the position of the snake by thick lines.
An edge has two sides (vertices) and the snake takes the central route at the vertex to which the edge is directed.


To reverse an edge, the snake first clears the center route and then takes the center route once its head reaches the opposite vertex.

We embed vertices of the constraint graph as shown below, where each of the three colored parts is a half of an edge gadget shown above. The points where the edges touch make sure that the incoming weight of each vertex is at least $2$ by forcing the snake to take the center route of some edge gadgets.




Finally, the black lines of all edge gadgets are connected to form a single cycle, so the head of the snake chases its tail.
If between two edge gadgets, we make the black path sufficiently long, the snake must always traverse the black paths in the same cyclic order.

To show that the black paths can always be constructed in a planar manner, consider a spanning subtree (thick edges in the figure below) of the constraint graph.
Then we can make black edges follow this tree as illustrated below, resulting in a planar graph.




The target position of the snake can be derived by the same transformation.
Hence, reconfiguring a snake is PSPACE complete, even on planar graphs.