  In the synchrotron radiation model, the polarization property depends on both
the configuration of the magnetic field and the geometry of the visible
emitting region. Some peculiar behaviors in the X-ray afterglows of {\it Swift}
gamma-ray bursts (GRBs), such as energetic flares and the plateau followed by a
sharp drop, might by highly linearly-polarized because the outflows powering
these behaviors may be Poynting-flux dominated. Furthermore, the broken-down of
the symmetry of the visible emitting region may be hiding in current X-ray data
and will give rise to interesting polarization signatures. In this work we
focus on the polarization accompanying the very early sharp decline of GRB
X-ray afterglows. We show that strong polarization evolution is possible in
both the high latitude emission model and the dying central engine model which
are used to interpret this sharp X-ray decline. It is thus not easy to
efficiently probe the physical origin of the very early X-ray sharp decline
with future polarimetry. Strong polarization evolution is also possible in the
decline phase of X-ray flares and in the shallow decline phase of X-ray light
curves characterized by chromatic X-ray VS. Optical breaks. An {\it XRT}-like
detector but with polarization capability on board a {\em Swift}-like satellite
would be suitable to test our predictions.
