  Two-dimensional conformal field theories (CFTs) defined on non-orientable
Riemann surfaces obey consistency Cardy conditions analogous to those in the
orientable case. We revisit those conditions for irrational theories with
central charge $c>1$ in the context of two-point functions of primaries on the
Real Projective plane $\mathbb{RP}^2$ and the partition function on the Klein
bottle $\mathbb{K}^2$. Using the irrational versions of the Virasoro fusion and
modular kernels we derive universal expressions for the non-orientable CFT data
at large conformal dimension, assuming a gap in the spectrum of scalar
primaries. In particular, we derive asymptotic formulas at finite central
charge for the averaged Light-Light-Heavy product $C_{LLH}\times\Gamma_{H}$ of
OPE coefficients with the $\mathbb{RP}^2$ one-point function normalizations, as
well as for the parity-weighted density of heavy scalar primaries (or
equivalently the density of heavy $\Gamma_H^2$). We discuss the gravitational
interpretation of the results.
