In this contribution, both theoretical studies and measurement results are reported for investigating the anisotropic property of propagation channels with respect to (w.r.t.) the (linear) polarizations of the transmitter (Tx) and receiver (Rx) antennas. Under the assumption that the channel impulse response (CIR) is polarization isotropic, analytical expressions are derived for the K-factor and delay spread as functions of Tx-Rx polarizations, and furthermore, our deduction also shows that the rank of the signal subspace spanned by a multi-polarized narrowband channel is upper-bounded. Based on the measurements conducted in indoor scenarios, empirical CIRs measured with 12 Tx- and 2 Rx-polarization directions are applied to investigating the channel polarization anisotropy. It can be observed that the measured channels exhibit properties inconsistent with the theoretical results deduced under the polarization-isotropy assumption, implying that the realistic indoor propagation channels can be anisotropic w.r.t. the Tx-Rx polarizations.