A few years back, I've been told by a tyre salesmen that directional tyres need rotation more often than non-directional ones, like every 5k miles or sooner, since they can only go back and forth, so you must make sure they have more even wear, since you can't switch left-to-right. (Mind you, it was at an America's Tire, that does tyre rotation free of charge.)
However, as per discussion at Does more frequent tyre rotation make tyres last any longer?, it would appear that such uneven-axle-wear property -- that you can't really do much about differential imperfection -- would actually appear to make the frequent rotation more useful to non-directional tyres, since you'd probably want each tyre to travel on each wheel during its lifetime equally, whereas on directional tyres nothing like that is possible.
As such, it would seem like in perfect conditions, at least 3 rotations are warranted for non-directional tyres, and at least 1 for directional ones.
I would guesstimate that in practice, since it's difficult to predict how long the tyres will last, and the cycle with directional is so much shorter than with non-directional -- only 1 vs. 3 -- that directional tyres warranted to 60k really have no purpose being rotated prior to 15k to 20k, unless you really expect to get fewer than 30k or 40k miles out of it, respectively, whereas it's the non-directional ones that would benefit from the less-than-6k mile rotation the most.
Does this make sense, or am I missing something?