In this work, we provide a more complete picture on the relationship between submodularity with concavity by extending some of the results connecting submodularity with convexity to the concave aspects of submodular functions. We first show the existence of the superdifferentials and tight modular upper bounds of a submodular function. While it is hard to characterize this polyhedron, we obtain inner and outer bounds on the superdifferential along with certain specific supergradients that have been useful and practical in some machine learning applications (which we briefly survey). We also show connections between optimality conditions over the superdifferentials and submodular maximization, and show how forms of approximate optimality conditions translate into approximation factors for maximization. Lastly, we consider versions of the discrete separation theorem and the Fenchel duality theorem when seen from the concave point of view. Joint work with Rishabh Iyer (http://melodi.ee.washington.edu/~rkiyer/)