  We study reconfiguration of independent sets in interval graphs under the
token sliding rule. We show that if two independent sets of size $k$ are
reconfigurable in an $n$-vertex interval graph, then there is a reconfiguration
sequence of length $\mathcal{O}(k\cdot n^2)$. We also provide a construction in
which the shortest reconfiguration sequence is of length $\Omega(k^2\cdot n)$.
  As a counterpart to these results, we also establish that
$\textsf{Independent Set Reconfiguration}$ is PSPACE-hard on incomparability
graphs, of which interval graphs are a special case.
