  The optical spectra of vertically stacked MoSe$_2$/WSe$_2$ heterostructures
contain additional 'interlayer' excitonic peaks that are absent in the
individual monolayer materials and exhibit a significant spatial charge
separation in out-of-plane direction. Extending on a previous study, we used a
many-body perturbation theory approach to simulate and analyse the excitonic
spectra of MoSe$_2$/WSe$_2$ heterobilayers with three stacking orders,
considering both momentum-direct and momentum-indirect excitons. We find that
the small oscillator strengths and corresponding optical responses of the
interlayer excitons are significantly stacking-dependent and give rise to high
radiative lifetimes in the range of 5-200\,ns (at T=4\,K) for the 'bright'
interlayer excitons. Solving the finite-momentum Bethe-Salpeter Equation, we
predict that the lowest-energy excitation should be an indirect exciton over
the fundamental indirect band gap (K$\rightarrow$Q), with a binding energy of
220\,meV. However, in agreement with recent magneto-optics experiments and
previous theoretical studies, our simulations of the effective excitonic
Land\'e g-factors suggest that the low-energy momentum-indirect excitons are
not experimentally observed for MoSe$_2$/WSe$_2$ heterostructures. We further
reveal the existence of 'interlayer' C excitons with significant exciton
binding energies and optical oscillator strengths, which are analogous to the
prominent band nesting excitons in mono- and few-layer transition-metal
dichalcogenides.
