An exact quantum mechanical theory is developed to treat four-atom reactions of the type AB+CD<->(BCD+A, ACD+B), where the atoms are constrained to move in a plane. The theory makes use of an unbiased set of hyperspherical coordinates. A method is proposed for implementing the theory that exploits the potential optimized discrete variable representation. Application is made to the calculation of rovibrational state-to-state reaction probabilities for the reaction H-2+OH<->H2O+H, in which the length of the OH spectator bond is held fixed. The results show that a rotating bond approximation, in which the H-2 molecule is not allowed to rotate, gives good results for vibrationally selected reaction probabilities. The effect of reactant rotation and vibration on the reactivity and product distributions is discussed for the reactions H-2+OH-->H2O+H and H2O+H-->H-2+OH.