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Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T 4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T 4-configurations in $\mathbb{R}^{2\times 2}$ 
is very rich; in particular, their collection is open as a subset of $(\mathbb{R}^{2\times2})^{4}$ 
. Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T 4-configurations.
