If the tires are asymmetrical, the second method is used.
Be careful with the terminology. It is perfectly acceptable (and preferred) to cross-rotate asymmetrical tires.
Asymmetrical tires are tires that have a tread compound and/or pattern on the
outside edge that differs from the tread compound and/or pattern on the
inside edge. These tires must be mounted on the wheel with the marked outside of the tire facing the outside of the wheel. However, the tire/wheel combo can be mounted in any of the 4 positions on the car, and can rotate either direction. Many tires on the market these days are indeed asymmetrical.
Directional tires are tires that have a tread pattern where the tire is designed to rotate in only one direction. These tires can be mounted on the wheel in any orientation, but the tire/wheel combo must be mounted on the side of the car where the tire will rotate according to the marked direction on the tire. This tire/wheel combo can only be fitted to one side of the car, and must be rotated during rotations so that it stays on that side of the car. If there is a desire to rotate the tire to the other side of the car, this will require that the tire be unmounted from the wheel and remounted in the correct direction.
Tires that are both asymmetrical and directional do not exist. Manufacturers would have to make these tires in two different part numbers that would be specific for the right side or the left side of the car. That's too much trouble for the (very little) benefit.
In addition to the tire tread pattern and compound, some cars have different
sized tires on the rear axle than on the front axle. This is referred to as a
staggered setup. For staggered setups, rotations cannot be made between the rear axle and front axle due to the different sizes. The only possible rotation for a staggered setup is a same-axle swap (LR <-> RR, and LF <-> RF). To do this, the tires cannot be directional (they
can be asymmertrical). If a car is fitted with directional tires in a staggered setup, then no rotation is possible without dismounts and remounts of the tires, and this is not desirable.