This note investigates the boundary between polynomially-solvable Max Cut and NP Hard Max Cut instances when they are classified only on the basis of the sign pattern of the objective function coefficients, i.e., of the orthant containing the objective function vector. It turns out that the matching number of the subgraph induced by the positive edges is the key parameter that allows us to differentiate between polynomially-solvable and hard instances of the problem. We give some applications of the polynomially solvable cases.