  Dipolar (or spatially indirect) excitons (IXs) in semiconductor double
quantum well (DQW) subjected to an electric field are neutral species with a
dipole moment oriented perpendicular to the DQW plane. Here, we theoretically
study interactions between IXs in stacked DQW bilayers, where the dipolar
coupling can be either attractive or repulsive depending on the relative
positions of the particles. By using microscopic band structure calculations to
determine the electronic states forming the excitons, we show that the
attractive dipolar interaction between stacked IXs deforms their electronic
wave function, thereby increasing the inter-DQW interaction energy and making
the IX electrically polarizable. Many-particle effects interaction are
addressed by considering the coupling between a single IX in one of the DQWs to
a cloud of IXs in the other DQW, which is modeled either as a closed-packed
lattice or as a continuum IX fluid. We find that the lattice model yields IX
interlayer binding energies decreasing with increasing lattice density. This
behavior is due to the dominating role of the intra-DQW dipolar repulsion,
which prevents more than one exciton from entering the attractive region of the
inter-DQW coupling. Finally, both models shows that the single IX distorts the
distribution of IXs in the adjacent DQW, thus inducing the formation of an IX
polaron. While the interlayer binding energy reduces with IX density for
lattice polarons, the continuous polaron model predicts a non-monotonous
dependence on density in semi-quantitative agreement with a recent experimental
study [cf. Hubert {\it et al.}, Phys. Rev. {\bf X}9, 021026 (2019)].
