We study global attractors \(\mathcal {A}=\mathcal {A}_f\) of semiflows generated by semilinear partial parabolic differential equations of the form \(u_t = u_{xx} + f(x,u,u_x), 0<x<1\), satisfying Neumann boundary conditions. The equilibria \(v\in \mathcal {E}\subset \mathcal {A}\) of the semiflow are the stationary solutions of the PDE, hence they are solutions of the corresponding second order ODE boundary value problem. Assuming hyperbolicity of all equilibria, the dynamic decomposition of \(\mathcal {A}\) into unstable manifolds of equilibria provides a geometric and topological characterization of Sturm global attractors \(\mathcal {A}\) as finite regular signed CW-complexes, the Sturm complexes, with cells given by the unstable manifolds of equilibria. Concurrently, the permutation \(\sigma =\sigma _f\) derived from the ODE boundary value problem by ordering the equilibria according to their values at the boundaries \(x=0,1\), respectively, completely determines the Sturm global attractor \(\mathcal {A}\). Equivalently, we use a planar curve, the meander \(\mathcal {M}=\mathcal {M}_f\), associated to the the ODE boundary value problem by shooting. In the previous paper (Fiedler and Rocha in J Dyn Differ Equ, 2020. https://doi.org/10.1007/s10884-020-09836-5), we set up to determine the boundary neighbors of any specific unstable equilibrium \(\mathcal {O}\), based exclusively on the information on the corresponding signed hemisphere complex. In addition, a certain minimax property of the boundary neighbors was established. In the signed hemisphere decomposition of the cell boundary of \(\mathcal {O}\), this property identifies the equilibria which are closest to, or most distant from, \(\mathcal {O}\) at the boundaries \(x=0,1\), in each hemisphere. The main objective of the present paper is to derive this minimax property directly from the Sturm permutation \(\sigma \), or equivalently from the Sturm meander \(\mathcal {M}\), based on the Sturm nodal properties of the solutions of the ODE boundary value problem. This minimax result simplifies the task of identifying the equilibria on the cell boundary of each unstable equilibrium, directly from the Sturm meander \(\mathcal {M}\). We emphasize the local aspect of this result by an example for which the identification of the equilibria is obtained from the knowledge of only a segment of the Sturm meander \(\mathcal {M}\).