  Let $\mathrm{SO}^+(p,q)$ denote the identity connected component of the real
orthogonal group with signature $(p,q)$. We give a complete description of the
spaces of continuous and generalized translation- and
$\mathrm{SO}^+(p,q)$-invariant valuations, generalizing Hadwiger's
classification of Euclidean isometry-invariant valuations. As a result of
independent interest, we identify within the space of translation-invariant
valuations the class of Klain-Schneider continuous valuations, which strictly
contains all continuous translation-invariant valuations. The operations of
pull-back and push-forward by a linear map extend naturally to this class.
