I would not recommend cross rotating any radial tire.
The radial belts in a new tire, start off being "vertical". When the tire heats up during normal driving, the belts sort of creep a bit in one direction, then when they cool down, the rubber sets these belts in their new position...which is now not "vertical".
If you switch rotational direction on a radial, those same radial belts now have to creep back to vertical, then creep in the new direction.
This amount of movement within the tire can lead to the radial belts separating from the surrounding rubber holding the belt in place and can lead to bumps in the tread, or even having the entire tread separate.

These W126 respond very well to Michelin tires for some reason.