We investigate the polarization of emission from pairs of coupled whispering-gallery-mode semiconductor microdisks. The finite-difference time-domain method is used to carry out an electromagnetic field profile analysis of four classes of bonding and antibonding modes. For coupled circular microdisks, our simulations predict distinct in-plane polarization directions for the bonding and antibonding modes, in agreement with experimental measurements. For pairs of elliptical microdisks coupled along either the major or the minor axes, our simulations do not predict any polarized emission. Experimental measurements of elliptical microdisks coupled along the minor axis appear to be consistent with these predictions, with no preferred polarization direction for the bonding and antibonding modes. However, elliptical microdisks coupled along the major axis often show a preferred polarization at 45deg to the major axis, inconsistent with both the simulations and with simple symmetry considerations. This suggests that the polarization characteristics of the emission in elliptical geometries may be very sensitive to extrinsic symmetry-breaking factors such as patterning imperfections.