  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.
