  The game of peg solitaire on graphs was introduced by Beeler and Hoilman in
2011. In this game, pegs are initially placed on all but one vertex of a graph
$G$. If $xyz$ forms a path in $G$ and there are pegs on vertices $x$ and $y$
but not $z$, then a {\em jump} places a peg on $z$ and removes the pegs from
$x$ and $y$. A graph is called solvable if, for some configuration of pegs
occupying all but one vertex, some sequence of jumps leaves a single peg. We
study the game of {\em reversible peg solitaire}, where there are again
initially pegs on all but one vertex, but now both jumps and unjumps (the
reversal of a jump) are allowed. We show that in this game all non-star graphs
that contain a vertex of degree at least three are solvable, that cycles and
paths on $n$ vertices, where $n$ is divisible by $2$ or $3$, are solvable, and
that all other graphs are not solvable. We also classify the possible starting
hole and ending peg positions for solvable graphs.
