DNA-driven assembly of nanoscale objects has emerged as a powerful platform for the creation of materials by design via self-assembly. Recent years have seen much progress in the experimental realization of this approach for three-dimensional systems. In contrast, two-dimensional (2D) programmable nanoparticle (NP) systems are not well explored, in part due to the difficulties in creating such systems. Here we demonstrate the use of charged liquid interfaces for the assembly and reorganization of 2D systems of DNA-coated NPs. The absorption of DNA-coated NPs to the surface is controlled by the interaction between a positively charged lipid layer and the negatively charged DNA shells of particles. At the same time, interparticle interactions are switchable, from electrostatic repulsion between DNA shells to attraction driven by DNA complementarity, by increasing ionic strength. Using in situ surface X-ray scattering methods and ex situ electron microscopy, we reveal the corresponding structural transformation of the NP monolayer, from a hexagonally ordered 2D lattice to string-like clusters and finally to a weakly ordered network of DNA cross-linked particles. Moreover, we demonstrate that the ability to regulate 2D morphology yields control of the interfacial rheological properties of the NP membrane: from viscous to elastic. Theoretical modeling suggests that the structural adaptivity of interparticle DNA linkages plays a crucial role in the observed 2D transformation of DNA-NP systems at liquid interfaces.