In a combined theoretical and experimental work, we investigate X-ray absorption near-edge structure spectroscopy of the I L3 and the Pb M5 edges of the methylammonium lead iodide (MAPbI3) hybrid inorganic-organic perovskite and its binary phase PbI2. The absorption onsets are dominated by bound excitons with sizable binding energies of a few hundred millielectronvolts and pronounced anisotropy. The spectra of both materials exhibit remarkable similarities, suggesting that the fingerprints of core excitations in MAPbI3 are essentially given by its inorganic component, with negligible influence from the organic groups. The theoretical analysis complementing experimental observations provides the conceptual insights required for a full characterization of this complex material.