  Geometrical moir\'e patterns, generic for almost aligned bilayers of
two-dimensional (2D) crystals with similar lattice structure but slightly
different lattice constants, lead to zone folding and miniband formation for
electronic states. Here, we show that moir\'e superlattice (mSL) effects in
$\mathrm{MoSe}_2/\mathrm{WS}_2$ and $\mathrm{MoTe}_2/\mathrm{MoSe}_2$
heterobilayers that feature alignment of the band edges are enhanced by
resonant interlayer hybridization, and anticipate similar features in twisted
homobilayers of TMDs, including examples of narrow minibands close to the
actual band edges. Such hybridization determines the optical activity of
interlayer excitons in transition-metal dichalcogenide (TMD) heterostructures,
as well as energy shifts in the exciton spectrum. We show that the resonantly
hybridized exciton (hX) energy should display a sharp modulation as a function
of the interlayer twist angle, accompanied by additional spectral features
caused by umklapp electron-photon interactions with the mSL. We analyze the
appearance of resonantly enhanced mSL features in absorption and emission of
light by the interlayer exciton hybridization with both intralayer A and B
excitons in $\mathrm{MoSe}_2/\mathrm{WS}_2$, $\mathrm{MoTe}_2/\mathrm{MoSe}_2$,
$\mathrm{MoSe}_2/\mathrm{MoS}_2$, $\mathrm{WS}_2/\mathrm{MoS}_2$, and
$\mathrm{WSe}_2/\mathrm{MoSe}_2$.
