We relate the so-called powercone models of mixed non-deterministic and probabilistic choice proposed by Tix, Keimel, Plotkin, Mislove, Ouaknine, Worrell, Morgan, and McIver, to our own models of previsions. Under suitable topological assumptions, we show that they are isomorphic. We rely on Keimel's cone-theoretic variants of the classical Hahn-Banach separation theorems, using functional analytic methods, and on the Schröder-Simpson Theorem. Lemma 3.4 in the original 2017 version, published at MSCS, had a wrong proof, and we prove a repaired, albeit slightly less general version here.