Nonadiabatic coupling terms (NACTs) between different electronic states lead to fast radiationless decay in photoexcited molecules. Using molecular symmetry, i.e., symmetry with respect to permutation of identical nuclei and inversion of the molecule in space, the irreducible representations of the NACTs can be determined with a combination of molecular symmetry arguments and quantization rules. Here, we extend these symmetry rules for electronic states and coupling elements and demonstrate the importance of molecular symmetry for nonadiabatic nuclear dynamics. As an example, we consider the NACTs related to the torsion around the CN bond in C5H4NH. We present the results of quantum dynamical simulations of the photoinduced large amplitude torsion on three coupled electronic states and show how the interference between wavepackets leads to radiationless decay, which depends on the symmetry of the NACTs. Moreover, we show that the nuclear spin of the system determines the symmetry of the initial nuclear wave function and thus influences the torsional dynamics. This may open new possibilities for nuclear spin selective laser control of nuclear dynamics.