Recently the authors showed that there is a robust potential theory attached to any calibrated manifold $(X, ø). In particular, on X there exist ø -plurisubharmonic functions, ø-convex domains, ø-convex boundaries, etc., all inter-related and having a number of good properties. In this paper we show that, in a strong sense, the plurisubharmonic functions are the polar duals of the ø-submanifolds, or more generally, the ø currents studied in the original paper on calibrations. In particular, we establish an analogue of Duval-Sibony Duality which characterizes points in the ø-convex hull of a compact set K C X in terms of ø-positive Green's currents on X and Jensen measures on K. We also characterize boundaries of ø currents entirely in terms of ø-plurisubharmonic functions. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.