  We examine the effects of the axion coupling to $R\tilde{R}$ on the evolution
of cosmic structures. It is shown that the evolutions of the scalar- and
vector-type perturbations are not affected by this axion coupling. However the
axion coupling causes an asymmetric evolution of the two polarization states of
the tensor-type perturbation, which may lead to a sizable polarization
asymmetry in the cosmological gravitational wave if inflation involves a period
in which the axion coupling is important. The polarization asymmetry produced
during inflation are conserved over the subsequent evolution as long as the
scales remain in the large-scale limit, and thus this may lead to an observable
trace in the cosmic microwave background radiation.
