  Let $\Lambda_1$, $\Lambda_2$ be two discrete orbits under the linear action
of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ on the Euclidean plane. We
prove a Siegel$-$Veech-type integral formula for the averages $$
\sum_{\mathbf{x}\in\Lambda_1} \sum_{\mathbf{y}\in\Lambda_2} f(\mathbf{x},
\mathbf{y}) $$ from which we derive new results for the set $S_M$ of holonomy
vectors of saddle connections of a Veech surface $M$. This includes an
effective count for generic Borel sets with respect to linear transformations,
and upper bounds on the number of pairs in $S_M$ with bounded determinant and
on the number of pairs in $S_M$ with bounded distance. This last estimate is
used in the appendix to prove that for almost every $(\theta,\psi)\in S^1\times
S^1$ the translations flows $F_\theta^t$ and $F_\psi^t$ on any Veech surface
$M$ are disjoint.
