We present a semiclassical study of the rovibronic Renner-Teller (RT) effect on the reaction N(D-2) + H-2(X(1)Sigma(g)(+)) --> NH2(X(2)A" + A(2)A') --> NH(X(3)Sigma(-) + a(1)Delta) + H(S-2). A new approach for describing RT coupling within the framework of trajectory surface hopping is developed for this application. The calculations include the two lowest-lying electronic states of NH2 as derived from ab initio calculations together with a semiclassical description of the RT coupling. We investigated the role of the RT effect on the integral and differential reactive cross sections for formation of both NH electronic states and on the product distribution by varying the collision energy and the initial rotational state of H-2. Our results show that the RT coupling does not affect the reactivity of trajectories starting on the X(2)A" potential but considerably affects those starting on A(2)A', leading to an almost exclusive formation of NH(X(3)Sigma(-)) at collision energies below 5 kcal/mol and a maximum contribution of similar to10% to the overall formation of NH(X(3)Sigma(-)) for collision energies between 5 and 10 kcal/mol. The dependence of the RT effect on the collision energy and on the rovibrational state of H-2 has also been analyzed, and a simple classical interpretation of most of these features based on the time spent by trajectories in the region where the RT coupling is active is proposed. Our results predict that the RT effect is not important to the thermal rate constant but has measurable consequences for certain state-resolved experiments.