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?
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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.
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$\begingroup$ >I do not know what you mean when you say "The moon-earth axis". : Perhaps what was meant was the barycentre of the Earth-Moon system? $\endgroup$– jimCommented Jan 31, 2018 at 18:01
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$\begingroup$ The barycenter is only a point. The rotational axis for the two-body system that is the Earth and Moon would pass through that, yes, but that axis would be perpendicular to the Moon's Orbital plane. The Moon's rotational axis is not perpendicular to that plane. So the answer is: no, the Moon's rotation is not such that the Moon's direction in relation to that axis is constant. $\endgroup$– MichaelKCommented Feb 2, 2018 at 10:50