  We consider the Whitney problem for valuations: does a smooth $j$-homogeneous
translation-invariant valuation on $\mathbb R^n$ exist that has given
restrictions to a fixed family $S$ of linear subspaces? A necessary condition
is compatibility: the given valuations must coincide on intersections. We show
that for $S=\mathrm{Gr}_r(\mathbb R^n)$, the grassmannian of $r$-planes, this
condition becomes sufficient once $r\geq j+2$. This complements the Klain and
Schneider uniqueness theorems with an existence statement, and provides a
recursive description of the image of the cosine transform. Informally
speaking, we show that the transition from densities to valuations is localized
to codimension $2$.
  We then look for conditions on $S$ when compatibility is also sufficient for
extensibility, in two distinct regimes: finite arrangements of subspaces, and
compact submanifolds of the grassmannian. In both regimes we find unexpected
flexibility. As a consequence of the submanifold regime, we prove a Nash-type
theorem for valuations on compact manifolds, from which in turn we deduce the
existence of Crofton formulas for all smooth valuations on manifolds. As an
intermediate step of independent interest, we construct Crofton formulas for
all odd translation-invariant valuations.
