Relative state-to-state cross sections are obtained for OH colliding with CO, N-2, and CO2. Hexapole state selection is used to prepare a beam of OH molecules in the upper h-doublet component of the lowest rotational state. The collision induced rotational energy transfer is monitored state selectively by means of LIF (laser induced fluorescence) spectroscopy. A study is made of the symmetry effects in the obtained cross sections. The results are compared with previously reported cross sections for Ar, He, and H-2 A general propensity is found for parity conserving transitions to the II3/2, J=5/2 and II1/2, J=1/2 states.