  We develop the many-body theory of dipolar exciton-polaritons in an optical
microcavity in crossed transverse electric and in-plane magnetic fields. Even
for relatively weak fields, we reveal the existence of two minima in the bare
lower-polariton dispersion, which give rise to the tuneable transition between
the polariton Bose-Einstein condensate and that of excitons, produced by the
competition between these minima. We predict that such dipolar condensate
exhibits a roton-maxon character of the excitation spectrum, never before
observed for polaritons. We show that upon the transition between the two
condensation regimes, the weak correlations in the polariton gas give way to
the intermediate interparticle correlations characteristic for excitons, and
that the transition is accompanied by a sharp quenching of photoluminescence as
the lifetime is increased by several orders of magnitude. While in the
polariton regime, the luminescence peak corresponding to the condensate is
shifted to a non-zero angle. The angular dependence of the two-photon decay
time in the Hanbury Brown and Twiss experiment is calculated and used as a tool
to evidence the formation of the macroscopically-coherent state. Our proposal
opens opportunities towards manipulating the superfluid properties and
extended-range dipole-dipole correlations of exciton-polariton condensates.
