Effect of the polarizations of two counter-propagating relativistic laser pulses interacting with subwavelength-thin solid-density foil is investigated. Three-dimensional particle-in-cell simulations and analytical modeling show that the interaction and resulting transverse instability depend strongly on the polarization direction as well as the intensity distribution of the resultant light field in the foil. The left and right circularly polarized laser pair having the same phase at the common focal spot in the ultrathin foil leads to strongest distortion of the foil. The fastest growing mode and maximum growth rate depend mainly on the laser intensity. For all polarization and phase-difference combinations, the instability is weakest when the two laser pulses are exactly out of phase at the common focus in the foil.