Spherical fullerenes offer noteworthy structures usually involving six- and five-membered faces, with application in technological issues. In this sense, cavernous spherical-like structures bearing larger holes provide interesting examples for further understanding of structure-properties relationship. Here, we explored the magnetic response of a proposed cavernous nitride fullerene, C24N24, which has a O-h-symmetry with six N-4-macrocyclic and eight 1,3,5-triazine faces displaying 48-pi electrons. C24N24 exhibits a local aromatic behavior owing to the contrasting antiaromatic response of the N-4-macrocyclic faces and the aromatic character of the 1,3,5-triazine faces. Thus, the overall structure is ascribed as a local aromatic species, where the triazine faces exhibit the characteristic shielding cone for aromatic rings. Furthermore, the constructive combination of local shielding cones in C24N24 delivers a related shielding-cone response, as expected for a perfect aromatic cage. Hence, the local aromatic/nonaromatic/antiaromatic sections exhibit an additive or subtractive interaction, leading to a characteristic response inherent to the nature of the spherical cage. We expect that further study of the interplay between different aromatic and antiaromatic faces in fullerene-like cages can deliver interesting pseudo-aromatic or pseudo-antiaromatic spherical species.