  We consider a high-Q microresonator with $\chi^{(2)}$ nonlinearity under
conditions when the coupling rates between the sidebands around the pump and
second harmonic exceed the damping rates, implying the strong coupling regime
(SC). Using the dressed-resonator approach we demonstrate that this regime
leads to the dominance of the Hermitian part of the operator driving the
side-band dynamics over its non-Hermitian part responsible for the parametric
gain. This has allowed us to introduce and apply the cross-area concept of the
polariton quasi-particles and define their effective masses in the context of
$\chi^{(2)}$ ring-microresonators. We further use polaritons to predict the
modified spectral response of the resonator to a weak probe field, and to
reveal splitting of the bare-resonator resonances, avoided crossings, and Rabi
dynamics. Polariton basis also allows deriving a discrete sequence of the
parametric thresholds for the generation of sidebands of different orders.
