  We study the integrable crosscap states of the integrable quantum spin chains
and we classify them for the $\mathfrak{gl}(N)$ symmetric models. We also give
a derivation for the exact overlaps between the integrable crosscap states and
the Bethe states. The first part of the derivation is to calculate sum formula
for the off-shell overlap. Using this formula we prove that the normalized
overlaps of the multi-particle states are ratios of the Gaudin-like
determinants. Furthermore we collect the integrable crosscap states which can
be relevant in the AdS/CFT correspondence.
