The multistate vibronic interactions in the benzene radical cation are investigated theoretically, employing the framework of a linear vibronic coupling scheme. The five lowest electronic states are included in the treatment; in view of the degeneracy of some states, this amounts to eight coupled potential energy surfaces. Different types of ab initio calculations have been performed for the system parameters and been found to be in good mutual agreement, thus supporting each other. The calculations reveal a whole sequence of low-energy conical intersections between the potential energy surfaces of different states. Their importance for the nuclear dynamics in this prototypical organic radical cation is pointed out. Wave-packet dynamical simulations for these coupled potential energy surfaces will be presented in the following paper (Paper II).