  We introduce the poset of biflats of a matroid $M$, a Lagrangian analog of
the lattice of flats of $M$, and study the topology of its order complex, which
we call the biflats complex. This work continues the study of the Lagrangian
combinatorics of matroids, which was recently initiated by work of Ardila,
Denham and Huh. We show the biflats complex contains two distinguished
subcomplexes: the conormal complex of $M$ and the simplicial join of the
Bergman complexes of $M$ and $M^\perp$, the matroidal dual of $M$. Our main
theorems give sequences of elementary collapses of the biflats complex onto the
conormal complex and the join of the Bergman complexes of $M$ and $M^\perp$.
These collapses give a combinatorial proof that the biflats complex, conormal
complex and the join of the Bergman complexes of $M$ and $M^\perp$ are all
simple homotopy equivalent. Although simple homotopy equivalent, these
complexes have many different combinatorial properties. We collect and prove a
list of such properties.
