  Let $G=(V,E)$ be a finite undirected graph. Orient the edges of $G$ in an
arbitrary way. A $2$-cycle on $G$ is a function $d : E^2\to \mathbb{Z}$ such
for each edge $e$, $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$, and
$d(e, f) = 0$ whenever $e$ and $f$ have a common vertex. We show that each
$2$-cycle is a sum of three special types of $2$-cycles: cycle-pair $2$-cycles,
Kuratowski $2$-cycles, and quad $2$-cycles. In case that the graph is
Kuratowski connected, we show that each $2$-cycle is a sum of cycle-pair
$2$-cycles and at most one Kuratowski $2$-cycle. Furthermore, if $G$ is
Kuratowski connected, we characterize when every Kuratowski $2$-cycle is a sum
of cycle-pair $2$-cycles. A $2$-cycles $d$ on $G$ is skew-symmetric if $d(e,f)
= -d(f,e)$ for all edges $e,f\in E$. We show that each $2$-cycle is a sum of
two special types of skew-symmetric $2$-cycles: skew-symmetric cycle-pair
$2$-cycles and skew-symmetric quad $2$-cycles. In case that the graph is
Kuratowski connected, we show that each skew-symmetric $2$-cycle is a sum of
skew-symmetric cycle-pair $2$-cycles. Similar results like this had previously
been obtained by one of the authors for symmetric $2$-cycles. Symmetric
$2$-cycles are $2$-cycles $d$ such that $d(e,f)=d(f,e)$ for all edges $e,f\in
E$.
