  Hybrid organic-inorganic perovskites have emerged as very promising materials
for photonic applications, thanks to the great synthetic versatility that
allows to tune their optical properties. In the two-dimensional (2D)
crystalline form, these materials behave as multiple quantum-well
heterostructures with stable excitonic resonances up to room temperature. In
this work strong light-matter coupling in 2D perovskite single-crystal flakes
is observed, and the polarization-dependent exciton-polariton response is used
to disclose new excitonic features. For the first time, an out-of-plane
component of the excitons is observed, unexpected for such 2D systems and
completely absent in other layered materials, such as transition-metal
dichalcogenides. By comparing different hybrid perovskites with the same
inorganic layer but different organic interlayers, it is shown how the nature
of the organic ligands controllably affects the out-of-plane exciton-photon
coupling. Such vertical dipole coupling is particularly sought in those
systems, e.g. plasmonic nanocavities, in which the direction of the field is
usually orthogonal to the material sheet. Organic interlayers are shown to
affect also the strong birefringence associated to the layered structure, which
is exploited in this work to completely rotate the linear polarization degree
in only few microns of propagation, akin to what happens in metamaterials.
