For what it's worth, here's a description of "Directional" radial tires:
QUOTE
A pair of radial tires consist of two tires each having a tread comprising annular land portions formed by a plurality of circumferential main grooves extending in the circumferential direction of the tire, and rows of blocks formed by extending transverse main grooves inclinedly in the circumferential direction in at least land portions located at axially opposite side edge portions of the tire. Tread patterns of these two tires are asymmetrical and have directional property. Transverse main grooves of each of the tires, which define the rows of the blocks to be located axially inside, are extended inclinedly downwardly toward the inner side of the tread in the width direction as viewed from the front side of the tire. Each of the transverse main grooves of the tire, which define the row of the blocks located axially on the outer side is extended in an almost inverse V-letter shape as viewed from the front side of the tire.
Cross rotation concerns with: different camber settings resulting in unusual wear patterns, shifting belts, separation of belts because of air bubbles created, vibrations, blow outs, were prevalent with older radial tires. I'm not sure the manufacturers have the same concerns with today's modern radial. However, some high performance tires do come with directional arrows embossed on the sidewall indicating in which direction they should be rotating.
Nevertheless, I would agree with Steverino, 1990's "favorite person on this forum" in that, if the tire has been designated as "directional" and the alignment is spot on, then the tires should be rotated on the same side if for no other reason than the tread blocks are likely designed for maximum water dispersal in only one direction. Although, it could be interesting to see what happens when mixing sizes as well as sides?

Hell, you can even change direction of tires without cross rotating them!