  For valuations on convex bodies in Euclidean spaces, there is by now a long
series of characterization and classification theorems. The classical template
is Hadwiger's theorem, saying that every rigid motion invariant, continuous,
real-valued valuation on convex bodies in $\mathbb{R}^n$ is a linear
combination of the intrinsic volumes. For tensor-valued valuations, under the
assumptions of isometry covariance and continuity, there is a similar
classification theorem, due to Alesker. Also for the local extensions of the
intrinsic volumes, the support, curvature and area measures, there are
analogous characterization results, with continuity replaced by weak
continuity, and involving an additional assumption of local determination. The
present authors have recently obtained a corresponding characterization result
for local tensor valuations, or tensor-valued support measures (generalized
curvature measures), of convex bodies in $\mathbb{R}^n$. The covariance assumed
there was with respect to the group ${\rm O}(n)$ of orthogonal transformations.
This was suggested by Alesker's observation, according to which in dimensions
$n> 2$, the weaker assumption of ${\rm SO}(n)$ covariance does not yield more
tensor valuations. However, for tensor-valued support measures, the distinction
between proper and improper rotations does make a difference. The present paper
considers, therefore, the local tensor valuations sharing the previously
assumed properties, but with ${\rm O}(n)$ covariance replaced by ${\rm SO}(n)$
covariance, and provides a complete classification. New tensor valued support
measures appear only in dimensions two and three.
