The answer depends on what disturbs the helicopter (with no disturbance it has no tendency). For example, let's say there's a gust of headwind, blowing into the nose of the aircraft, increasing the airspeed. This adds more lift to the advancing side of the main rotor (pilot's right side for CCW), which has the effect of flapping the rotor up 90 degrees later over the nose and pitching the helicopter nose up. This end effect (pitch up) is the same regardless of the direction of rotor rotation.
Depending on the rotor design, the largest increase in flapping may be less than 90 degrees, which would add a (very secondary) roll effect. Here the direction of rotation matters. For a CCW main rotor this would cause a "left wing down" roll moment.
For more about this 90 degree (or less for some rotor designs) flapping offset see this: Basic Helicopter Flap Dynamics.
Another effect of the increased airspeed is that it focuses more of the wake in the rear of the rotor, decreasing lift there relative to the front of the rotor. This causes the rotor to flap down 90 degrees later on the pilot's right side (CCW), which adds a "right wing down" roll moment.
Analogous comments apply to the right wing down bank turn. Let's say the pilot hadn't trimmed the climb rate yet, he entered the turn but is now accelerating down due to the main rotor thrust being tilted further from vertical. Since the rotor is tilted right side down, the increase in air flow up (due to the helicopter falling down faster) adds velocity and lift to the rear of the rotor (CCW) in the same way increased airspeed added lift to the right side of the rotor. This lift will cause the right side of the rotor to flap up, pushing back towards a wings-level condition.
As you can see, in both cases, the primary response is stabilizing. This is described better here: Helicopter Stability