  We continue our study of the reflectionless measures associated to an
$s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting in $\mathbb{R}^d$
with $s\in (0,d)$. Here, our focus will be the study of CZOs that are rigid, in
the sense that they have few reflectionless measures associated to them. Our
goal is to prove that the rigidity properties of a CZO $T$ impose strong
geometric conditions upon the support of any measure $\mu$ for which $T$ is a
bounded operator in $L^2(\mu)$. In this way, we shall reduce certain well-known
problems at the interface of harmonic analysis and geometric measure theory to
a description of reflectionless measures of singular integral operators. What's
more, we show that this approach yields promising new results.
