  The polarization of the 21-cm radiation from the epoch of reionization arises
from Thomson scattering of 21-cm photons from free electrons and provides
information that complements that from the intensity fluctuation. Previous work
showed that a direct detection of this signal will be difficult, and hinted
that the signal might be enhanced via correlation with other tracers. Here, we
discuss the cross-correlation between the cosmic microwave background (CMB)
polarization and the 21-cm polarization. We treat reionization using an
analytical model with parameters calibrated by semi-numerical simulations. We
then derive the cross-correlation angular power spectrum using the
total-angular-momentum formalism. We also provide a noise analysis to test
against two closely related, but subtly different, null hypotheses. First, we
assume no reionization as a null hypothesis, and determine how well this null
hypothesis could be ruled out by an observed 21cm-CMB polarization correlation.
Second, we determine how well the null hypothesis of no 21-cm polarization can
be ruled out by seeking the cross-correlation, assuming reionization is
established from the CMB. We find that the first question could be answered by
a synergy of ambitious next-generation 21-cm and CMB missions, whereas the
second question will still remain out of reach.
