The polarization characteristics of the indoor ultrawideband channel are investigated using dual-polar 3.1-10.6 GHz channel measurements. Under the generalized elliptical polarization framework, the channel cross-polar discrimination (XPD) statistics are evaluated. Lognormal and triangular distributions are shown to well model the XPD and ellipticity angle. Considerable spectral and small-scale spatial variability is observed in the XPD. Asymptotic convergence of XPD to its mean with increasing bandwidth is established. An analysis of the variation of tap XPD with excess delay indicates that the XPD is highest for the initial paths. It is shown that increased scattering leads to greater depolarization. The dependence of XPD on the channel's rms delay spread is modelled using a linear regression with -0.5 dB/ns slope. This characterization will aid the analysis of polarization-dependent loss and the design of polarized multiple-antenna systems.