Inverse melting or disordering, in which the disordered phase forms upon cooling, is known for a few cases in bulk systems under high pressure. We show that inverse disordering also occurs in two dimensions: For a monolayer of 1,4,5,8-naphthalene-tetracarboxylic dianhydride on Ag(111), a completely reversible order-disorder transition appears upon cooling. The transition is driven by strongly anisotropic interactions within the layer versus with the metal substrate. Spectroscopic data reveal changes in the electronic structure of the system corresponding to a strengthening of the interface bonding at low temperatures. We demonstrate that the delicate, temperature-dependent balance between the vertical and lateral forces is the key to understanding this unconventional phase transition.