Measures of concordance determined by $D_4$-invariant measures on $(0,1)^2$

A measure, $\mu$, on $(0,1)^2$ is said to be $D_4$-invariant if its value for any Borel set is invariant with respect to the symmetries of the unit square. A function, $\kappa$, generated in a certain way by a measure, $\mu$, on $(0,1)^2$ is shown to be a measure of concordance if and only if the generating measure is positive, regular, $D_4$-invariant, and satisfies certain inequalities. The construction examined here includes Blomqvist's beta as a special case.