We have studied the properties of the prototype hybrid organic-inorganic perovskite CH3NH3PbI3 using relativistic density functional theory (DFT). For our analysis we introduce the concept of CH3NH3+ "pair modes," that is, characteristic relative orientations of two neighboring CH3NH3+ cations. In our previous work [Li and Rinke, Phys. Rev. B 94, 045201 (2016)2469-995010.1103/PhysRevB.94.045201] we identified two preferential orientations that a single CH3NH3+ cation adopts in a unit cell. The total number of relevant pairs can be reduced from the resulting 196 combinations to only 25 by applying symmetry operations. DFT results of several 2×2×2 supercell models reveal the dependence of the total energy, band gap, and band structure on the distribution of CH3NH3+ cations and the pair modes. We have then analyzed the pair-mode distribution of a series of 4×4×4 supercell models with disordered CH3NH3+ cations. Our results show that diagonally oriented CH3NH3+ cations are rare in optimized CH3NH3PbI3 supercell structures. In the prevailing pair modes, the C-N bonds of the two neighboring CH3NH3+ cations are aligned approximately vertically. Furthermore, we fit the coefficients of a pair-mode expansion to our supercell DFT reference structures. The pair-mode model can then be used to quickly estimate the energies of disordered perovskite structures. Our pair-mode concept provides combined atomistic-statistical insight into disordered structures in bulk hybrid perovskite materials.