  Twisted bilayers of transition metal dichalcogenides (TMDs) have revealed a
rich exciton landscape including hybrid excitons and spatially trapped moir\'e
excitons that dominate the optical response of the material. Recent studies
have shown that in the low-twist-angle regime, the lattice undergoes a
significant relaxation in order to minimize local stacking energies. Here,
large domains of low energy stacking configurations emerge, deforming the
crystal lattices via strain and consequently impacting the electronic band
structure. However, so far the direct impact of atomic reconstruction on the
exciton energy landscape and the optical properties has not been well
understood. Here, we apply a microscopic and material-specific approach and
predict a significant change in the potential depth for moir\'e excitons in a
reconstructed lattice, with the most drastic change occurring in naturally
stacked TMD homobilayers. We show the appearance of multiple flat bands and a
significant change in the position of trapping sites compared to the rigid
lattice. Most importantly, we predict a multi-peak structure emerging in
optical absorption of WSe$_2$ homobilayers - in contrast to the single peak
that dominates the rigid lattice. This finding can be exploited as an
unambiguous signature of atomic reconstruction in optical spectra of moir\'e
excitons in naturally stacked twisted homobilayers.
