Knowledge of triplet state energies and triplet-triplet (T-T) interactions in aggregated organic molecules is essential for understanding photochemistry and dynamics of many natural and artificial systems. In this work, we combine direct phosphorescence measurements of triplet state energies, which are challenging due to the spin-forbidden nature of respective transitions and applicable to only a limited number of systems, with quantum chemical computational tools that can provide valuable qualitative and quantitative information about triplet states of interacting molecules. Using hexatriene, protoporphyrin, pheophorbide, and chlorophyll dimers as model systems, we demonstrate a complicated dependence of T-T coupling on a relative orientation of chromophores, governed by a nodal structure of overlapping electronic wave functions, that modulates interpigment interactions by orders of magnitude. It is also shown that geometrical relaxation of the triplet state is one of the critical factors for predictive modeling of T-T interactions in molecular aggregates.