In this talk I will introduce the combinatorial class of rigid quadrangulations, which form a subclass of flat quadrangulations of the disk, meaning that all non-boundary vertices are of degree 4. Rigid quadrangulations are shown to be in bijection with certain integer-labeled quadrangulations of the sphere, that were enumerated recently by Bousquet-Mélou and Elvey Price. The bijection relates several natural statistics on one side to equally natural, but rather different, statistics on the other. Finally, I will touch upon the question of scaling limits of large rigid quadrangulations and similar models of random flat metrics on the disk, and their physics motivation.