It is well known that the moon always shows the same side to the earth. But does the orientation of this side change? I.e. does the moon rotate around the moon-earth axis?
Yes, it changes
Yes, the Moon changes its orientation in relation to the Earth very slightly. This is called Libration.
The picture above is only a very limited view, in reality the motion is even more complex. Check the Wikipedia page linked above for a great animation for all of the year 2013.
I do not know what you mean when you say "The moon-earth axis".
The Moon's rotation is around its rotational axis. That rotational axis is not aligned or at a perfect right angle with any other axis or plane that we know of: not the Earth's rotational axis, nor the Ecliptic plane, nor the Moon's orbital plane.
Nor are the axises of the Moon's orbit aligned with any other rotational axis.