The combinatorial class of rigid quadrangulations is a subclass of theflat quadrangulations of the disk, meaning that all non-boundaryvertices are of degree 4. The uniform random rigid quadrangulation offixed perimeter is a natural discrete model of flat JT gravity, in thesense discussed by Ferrari in another talk. We show that rigidquadrangulations are in bijection with certain integer-labeledquadrangulations of the sphere, that were previously enumerated byBousquet-Mélou and Elvey Price.