  The decomposition of the space of continuous and translation invariant
valuations into a sum of SO(n) irreducible subspaces is obtained. A
reformulation of this result in terms of a Hadwiger type theorem for continuous
translation invariant and SO(n)-equivariant tensor valuations is also given. As
an application, symmetry properties of rigid motion invariant and homogeneous
bivaluations are established and then used to prove new inequalities of
Brunn-Minkowski type for convex body valued valuations.
