Let $F$ be a 4-regular graph. Each circuit partition $P$ of $F$ has a corresponding touch-graph $Tch(P)$; the circuits in $P$ correspond to vertices of $Tch(P)$, and the vertices of $F$ correspond to edges of $Tch(P)$. We discuss the connection between modified versions of the interlacement matrix of an Euler system of $F$ and the cycle space of $Tch(P)$, over $GF(2)$ and $\mathbb{R}$.