  The rotor-router model on a graph describes a discrete-time walk accompanied
by the deterministic evolution of configurations of rotors randomly placed on
vertices of the graph. We prove the following property: if at some moment of
time, the rotors form a closed clockwise contour on the planar graph, then the
clockwise rotations of rotors generate a walk which enters into the contour at
some vertex $v$, performs a number of steps inside the contour so that the
contour formed by rotors becomes anti-clockwise, and then leaves the contour at
the same vertex $v$. This property generalizes the previously proved theorem
for the case when the rotor configuration inside the contour is a cycle-rooted
spanning tree, and all rotors inside the contour perform a full rotation. We
use the proven property for an analysis of the sub-diffusive behavior of the
rotor-router walk.
