No, in forward flight it is essentially flapback/forward.
If you add or reduce collective, you change the pitch by the same amount all round the disc.
In forward flight there is a difference in V squared between the advancing and retreating side so the increase/decrease in pitch has a bigger effect on the advancing side. This is the same effect as introducing forward speed to a helicopter in the still air hover without changing collective pitch.
Therefore when you lower the lever the disc flaps forward (usually giving a nose drop) and when you raise the lever the disc flaps back. There may be some roll effects as well if the phase lag of the rotor system isn't exactly 90 degrees.
It has a correct term which I can't remember but it is something like Pitch Instability caused by Collective application.
The horizontal stabilisers and other airframe bits only cause any changes when the rate of climb or descent is established.