Continuing earlier work by Székelyhidi, we describe the
topological and geometric structure of so-called
T4-configurations which are the most prominent examples of
nontrivial rank-one convex hulls. It turns out that the structure of
T4-configurations in $\mathbb{R}^{2\times 2}$ is very rich; in particular,
their collection is open as a subset of $(\mathbb{R}^{2\times
2})^{4}$. Moreover a previously purely algebraic criterion is
given a geometric interpretation. As a consequence, we sketch an
improved algorithm to detect T4-configurations.