The optical spectra of vertically stacked MoSe2/WSe2 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 MoSe2/WSe2 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→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 MoSe2/WSe2 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.