Before revision:  By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved Fr\'echet--Hoeffding bounds to bounds on the expectation of f(S).
After revision:  By means of this representation, we provide an integral characterization of orthant orders on the set of quasi-copulas which relates the improved Fr\'echet-Hoeffding bounds to bounds on the expectation of f(S).