Depends on the aircraft design (i.e. how much "weather cocking" tendency it has) but in your example of an anti-clockwise main rotor, hovering in a crosswind from the left side will increase the requirement for right yaw pedal / tend towards a negative tail rotor pitch input.
In autorotation, the tail rotor needs enough negative pitch, in addition to that required to overcome transmission drag, to allow the pilot to balance the aircraft in a turn to the right.