But if you go to some tire sites or repair manuals, the criss-cross rotation pattern is recommended or at least acceptable (i.e., left front to right rear, right front to left rear). That would mean it's OK to run them in different directions?

If it were not OK, wouldn't they somehow be marked as directional? Like true directional tires are?

I've heard that too about not switching directions on steel-belted radials, but is that now an urban legend?