  We investigate a Bose-Einstein condensate strongly coupled to an optical
cavity via a repulsive optical lattice. We detect a stable self-ordered phase
in this regime, and show that the atoms order through an antisymmetric coupling
to the P-band of the lattice, limiting the extent of the phase and changing the
geometry of the emergent density modulation. Furthermore, we find a
non-equilibrium phase with repeated intense bursts of the intra-cavity photon
number, indicating non-trivial driven-dissipative dynamics.
