We study the fundamental problem 2NASH of computing a Nash equilibrium (NE) point in bimatrix games. We start by proposing a novel characterization of the NE set, via a bijective map to the solution set of a parameterized quadratic program (NEQP), whose feasible space is the highly structured set of correlated equilibria (CE). This is, to our knowledge, the first characterization of the subset of CE points that are in “1–1” correspondence with the NE set of the game, and contributes to the quite lively discussion on the relation between the spaces of CE and NE points in a bimatrix game (e.g., [15], [26] and [33]). We proceed with studying a property of bimatrix games, which we call mutually concavity (MC), that assures polynomial-time tractability of 2NASH, due to the convexity of a proper parameterized quadratic program (either NEQP, or a parameterized variant of the Mangasarian & Stone formulation [23]) for a particular value of the parameter. We prove various characterizations of the MC-games, which eventually lead us to the conclusion that this class is equivalent to the class of strategically zero-sum (SZS) games of Moulin & Vial [25]. This gives an alternative explanation of the polynomial-time tractability of 2NASH for these games, not depending on the solvability of zero-sum games. Moreover, the recognition of the MC-property for an arbitrary game is much faster than the recognition SZS-property. This, along with the comparable time-complexity of linear programs and convex quadratic programs, leads us to a much faster algorithm for 2NASH in MC-games. We conclude our discussion with a comparison of MC-games (or, SZS-games) to kk-rank games, which are known to admit for 2NASH a FPTAS when kk is fixed [18], and a polynomial-time algorithm for k=1k=1 [2]. We finally explore some closeness properties under well-known NE set preserving transformations of bimatrix games.