  Crofton formulas on simply-connected Riemannian space forms allow to compute
the volumes, or more generally the Lipschitz-Killing curvature integrals of a
submanifold with corners, by integrating the Euler characteristic of its
intersection with all geodesic submanifolds. We develop a framework of Crofton
formulas with distributions replacing measures, which has in its core Alesker's
Radon transform on valuations. We then apply this framework, and our recent
Hadwiger-type classification, to compute explicit Crofton formulas for all
isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and
pseudohyperbolic spaces. We find that, in essence, a single measure which
depends analytically on the metric, gives rise to all those Crofton formulas
through its distributional boundary values at parts of the boundary
corresponding to the different indefinite signatures. In particular, the
Crofton formulas we obtain are formally independent of signature.
