In light of the recent measurement of the η → π0γγ decay by the KLOE-2 Collaboration, we have extended a previous analysis including vector- and scalar-meson exchange contributions using the vector meson dominance and linear sigma model frameworks, respectively, to incorporate the effects of the a2(1320) tensor meson within a chiral context. Although the individual contribution of the a2 is negligible, its destructive interference with the vector-meson resonances is found to be significant, representing approximately 18% of the total signal and substantially affecting the diphoton invariant-mass distribution, especially at low γγ invariant mass values. The result obtained for the corresponding branching ratio is approximately 5σ below the reported value of the particle data group, while it is in very good agreement with the KLOE-2 measurement. In contrast, the total contribution of the a2 is found to be negligible in η’ → π0γγ, as this process is completely dominated by the exchange of an on-shell ω vector resonance.