assume your Transverse wave is and electromagnetic plane wave and the longitudinal wave is sound and the propagation medium as the atmosphere.
Due to "Superposition" principle, any particle in atmosphere handle each of these two waves separately. but your question is " can each of these wave change the properties of the other one?"
Yes, it is possible both the waves can interfere and after multiple reflection guided waves are formed. The polarization vector of these two waves are completely different (by 90 degrees, K1x + K2y). A new wave will form due to resultant of these waves and yes, this new wave can preserve the properties of their mother waves but it will not happen always.