In this paper we study some properties of the distribution function of the random variable C(X,Y) when the copula of the random pair (X,Y) is M (respectively, W) - the copula for which each of X and Y is almost surely an increasing (respectively, decreasing) function of the other -, and C is any copula. We also study the distribution functions of M(X,Y) and W(X,Y) given that the joint distribution function of the random variables X and Y is any copula.
We introduce four families of semilinear copulas (i.e. copulas that are linear in at least one coordinate of any point of the unit square) of which the diagonal and opposite diagonal sections are given functions. For each of these families, we provide necessary and sufficient conditions under which given diagonal and opposite diagonal functions can be the diagonal and opposite diagonal sections of a semilinear copula belonging to that family. We focus particular attention on the family of orbital semilinear copulas, which are obtained by linear interpolation on segments connecting the diagonal and opposite diagonal of the unit square.