  (To appear in The American Statistician.) Distance covariance (Sz\'ekely,
Rizzo, and Bakirov, 2007) is a fascinating recent notion, which is popular as a
test for dependence of any type between random variables $X$ and $Y$. This
approach deserves to be touched upon in modern courses on mathematical
statistics. It makes use of distances of the type $|X-X'|$ and $|Y-Y'|$, where
$(X',Y')$ is an independent copy of $(X,Y)$. This raises natural questions
about independence of variables like $X-X'$ and $Y-Y'$, about the connection
between Cov$(|X-X'|,|Y-Y'|)$ and the covariance between doubly centered
distances, and about necessary and sufficient conditions for independence. We
show some basic results and present a new and nontechnical counterexample to a
common fallacy, which provides more insight. We also show some motivating
examples involving bivariate distributions and contingency tables, which can be
used as didactic material for introducing distance correlation.
