Rock-salt lead selenide nanocrystals can be used as building blocks for large scale square superlattices via two-dimensional assembly of nanocrystals at a liquid-air interface followed by oriented attachment. Here we report Scanning Tunneling Spectroscopy (STS) measurements of the local density of states of an atomically coherent superlattice with square geometry made from PbSe nanocrystals. Controlled annealing of the sample permits the imaging of a clean structure and to reproducibly probe the band gap and the valence hole and conduction electron states. The measured band gap and peak positions are compared to the results of optical spectroscopy and atomistic tight-binding calculations of the square superlattice band Electronic properties of atomically coherent square PbSe nanocrystal superlattice resolved by Scanning Tunne structure. In spite of the crystalline connections between nanocrystals that induce significant electronic couplings, the electronic structure of the superlattices remains very strongly influenced by the effects of disorder and variability.