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 \( {\mathbbm{RP}}^2 \) and the partition function on the Klein bottle \( {\mathbbm{K}}^2 \) . 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 CLLH × ΓH of OPE coefficients with the \( {\mathbbm{RP}}^2 \) 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.
