  In this paper we study the relationship between two fundamental regularity
properties of an $s$-dimensional Calder\'{o}n-Zygmund operator (CZO) acting on
a Borel measure $\mu$ in $\mathbb{R}^d$, with $s\in (0,d)$.
  In the classical case when $s=d$ and $\mu$ is equal to the Lebesgue measure,
Calder\'on and Zygmund showed that if a CZO is bounded in $L^2$ then the
principal value integral exists almost everywhere. However, there are by now
several examples showing that this implication may fail for lower-dimensional
kernels and measures, even when the CZO has a homogeneous kernel consisting of
spherical harmonics.
  We introduce sharp geometric conditions on $\mu$, in terms of certain scaled
transportation distances, which ensure that an extension of the
Calder\'{o}n-Zygmund theorem holds. These conditions are necessary and
sufficient in the cases of the Riesz transform and the Huovinen transform. Our
techniques build upon prior work by Mattila and Verdera, and incorporate the
machinery of symmetric measures, introduced to the area by Mattila.
