The present computational study explores the nature of spin singlet and triplet electronic excitations in π-stacked aggregates of perylene-3,4:9,10-bis(dicarboximide) (PDI) derivatives. Concretely, we focus on the slip-stacked aggregation motive in the crystal structure of tetraphenyl PDI. Our study relies on electronic structure calculations of molecules, dimers and oligomers at the DFT and TDDFT level, and the characterization of excited states in terms of local excitations (LE) and charge transfer (CT) states. We rationalize the role of inter-chromophore CT states in the lowest singlets and triplets of PDI aggregates in terms of excitonic couplings and diabatic contributions. In this case, LE/LE and LE/CT couplings are both strong, but while the former induce H-aggregation, the latter promotes the stabilization of the optical state (J-aggregation). Hence, the photophysics of tetraphenyl PDI emerge as the competition between these two interactions. Interestingly, CT terms constitute about half of the transition to optical states, but they barely contribute to the nature of dark transitions. In the singlet state, this can be rationalized by the relation between electron and hole couplings. Triplet excitons, despite holding strong superexchange interactions, present a much larger LE/CT energy gap than in the singlet, restraining LE/CT mixings. These properties can be sensibly modified upon molecular distortions that tune diabatic energies and couplings. The conclusions of our study provide a deep understanding of aggregation effects, in particular for the much less explored triplet excitons. Moreover, they can be extended to π-stacked aggregates of PDI derivatives and generalized to the case of conjugated organic chromophores.