  Given two torus invariant Weil divisors $D$ and $D'$ on a two-dimensional
cyclic quotient singularity $X$, the groups
$\mathop{Ext}\nolimits^i_{X}(\mathcal{O}(D),\mathcal{O}(D'))$, $i>0$, are
naturally $\mathbb{Z}^2$-graded. We interpret these groups via certain
combinatorial objects using methods from toric geometry. In particular, it is
enough to give a combinatorial description of the
$\mathop{Ext}\nolimits^1$-groups in the polyhedra of global sections of the
Weil divisors involved. Higher $\mathop{Ext}\nolimits^i$-groups are then
reduced to the case of $\mathop{Ext}\nolimits^1$ via a quiver. We use this
description to show that
$\mathop{Ext}\nolimits^1_{X}(\mathcal{O}(D),\mathcal{O}(K-D')) =
\mathop{Ext}\nolimits^1_{X}(\mathcal{O}(D'),\mathcal{O}(K-D))$, where $K$
denotes the canonical divisor on $X$. Furthermore, we show that
$\mathop{Ext}\nolimits^{i+2}_{X}(\mathcal{O}(D),\mathcal{O}(D'))$ is the Matlis
dual of $\mathop{Tor}\nolimits_{i}^{X}(\mathcal{O}(D),\mathcal{O}(D'))$.
