Molecular chlorine was photolyzed using circularly polarized radiation at 310 and 330 nm, and orientation moments of the chlorine-atom Cl(P-2(j)) photofragment distributions were measured by resonance enhanced multiphoton ionization using circularly polarized light with Doppler resolution. The product atoms were found to be strongly oriented in the laboratory as a result of both incoherent and coherent dissociation mechanisms, and the orientation moments contributed by each of these mechanisms have been separately measured. The experimental results can be explained by nonadiabatic transitions from the C (1)Pi (1u) state to higher states of Omega =1(u) symmetry, induced by radial derivative coupling. Ab initio calculations indicate strong Rosen-Zener-Demkov noncrossing-type radial derivative couplings between states of 1(u) symmetry. The observed angular distribution (beta parameter) indicates that 88% of Cl*(P-2(1/2)) fragments produced at 310 nm originate from a perpendicular transition to the C state. The orientation measurements suggest that 67 +/- 16% of Cl-35*(P-2(1/2)) atoms dissociate via the 1(u)((3)Sigma (+)(1u)) state, and 21 +/-6% dissociate via the 1(u)((3)Delta (1u)) state.