  Cir X-1 is a neutron star X-ray binary characterized by strong variations in
flux during its eccentric $\sim$16.6 days orbit. There are also strong
variations in the spectral state, and historically it has shown both atoll and
Z state properties. We observed the source with the Imaging X-ray Polarimetry
Explorer during two orbital segments, 6 days apart, for a total of 263~ks. We
find an X-ray polarization degree in these segments of $1.6\%\pm0.3\%$ and
$1.4\%\pm0.3\%$ at polarization angles of $37^\circ\pm5^\circ$ and
$-12^\circ\pm7^\circ$, respectively. Thus we observed a rotation of the
polarization angle by $49^\circ\pm8^\circ$ along the orbit. Because variations
of accretion flow, and then of the hardness ratio, are expected during the
orbit, we also studied the polarization binned in hardness ratio, and found the
polarization angle differing by $67^\circ\pm11^\circ$ between the lowest and
highest values of the hardness ratio. We discuss possible interpretations of
this result that could indicate a possible misalignment between the symmetry
axes of the accretion disk and the Comptonizing region caused by the
misalignment of the neutron star's angular momentum with respect to the orbital
one.
