  We study interactions between polaritons, arising when photons strongly
couple to collective excitations in an array of two-level atoms trapped in an
optical lattice inside a cavity. We consider two types of interactions between
atoms: Dipolar forces and atomic saturability, which ranges from hard-core
repulsion to Rydberg blockade. We show that, in spite of the underlying
repulsion in the subsystem of atomic excitations, saturability induces a
broadband bunching of photons for two-polariton scattering states. We interpret
this bunching as a result of interference, and trace it back to the mismatch of
the quantization volumes for atomic excitations and photons. We examine also
bound bipolaritonic states: These include states created by dipolar forces, as
well as a gap bipolariton, which forms solely due to saturability effects in
the atomic transition. Both types of bound states exhibit strong bunching in
the photonic component. We discuss the dependence of bunching on experimentally
relevant parameters.
