  We study effects of the particle-wake interactions on the dispersion and
polarization of dust lattice wave modes in two-dimensional plasma crystals.
Most notably, the wake-induced coupling between the modes causes the branches
to "attract" each other, and their polarizations become elliptical. Upon the
mode hybridization the major axes of the ellipses (remaining mutually
orthogonal) rotate by $45^\circ$. To demonstrate importance of the obtained
results for experiments, we plot spectral densities of the longitudinal and
transverse waves revealing distinct fingerprints of the mixed polarization.
Furthermore, we show that at strong coupling the hybrid mode is significantly
shifted towards smaller wave numbers, away from the border of the first
Brillouin zone (where the hybrid mode is localized for a weak coupling).
