Matroids with different configurations and the same $\mathcal{G}$-invariant

From the configuration of a matroid (which records the size and rank of the cyclic flats and the containments among them, but not the sets), one can compute several much-studied matroid invariants, including the Tutte polynomial and a newer, stronger invariant, the $\mathcal{G}$-invariant. To gauge how much additional information the configuration contains compared to these invariants, it is of interest to have methods for constructing matroids with different configurations but the same $\mathcal{G}$-invariant. We offer several such constructions along with tools for developing more.