We complement the recently introduced classes of lower and upper semilinear copulas by two new classes, called vertical and horizontal semilinear copulas, and characterize the corresponding class of diagonals. The new copulas are in essence asymmetric, with maximum asymmetry given by $1/16$. The only symmetric members turn out to be also lower and upper semilinear copulas, namely convex sums of $\Pi$ and $M$.