Does the Earth help stabilize changes in the moon's obliquity as well? We know that the moon helps stabilize changes in Earth's obliquity.
But what about Earth and the moon? Are some of the obliquity-stabilizing effects (of the moon on the Earth) communicated through tidal dissipation (the friction which is presumably isotropic) - effects that would otherwise change the Earth's obliquity if it didn't experience that tidal dissipation from the moon?
 A: The Moon is currently stabilizing the Earth's obliquity to within a small range (22.1 to 24.5 degrees). Without this, the Earth's obliquity would be much more chaotic over long periods of time owing to resonances with other planets, especially Jupiter. For climate, the present obliquity of the Earth is more or less "just right", as per the Goldilocks Principle. No obliquity would result in permanent ice sheets covering the Earth's temporate zones. Very high obliquity results in extreme seasonal variation, from scorching summers to supercold winters.
The Moon's obliquity is about 6.5 degrees, but is locked to the orbital inclination to the ecliptic of 5 degrees, so that the Sun can rise no more than 1.5 degrees above the horizon at the lunar poles. It is important that this locking, called a Cassini state, stay for long periods in order to allow water ice and other volatiles to collect in shadowed crater floors near the lunar poles. If the Moon's obliquity had ever been unlocked in the past, these volatiles would have quickly escaped into space. See Effects of orbital evolution on lunar ice stability (Siegler, Bills, and Paige. Journal of Geophysical Research) for more (much more!).
Since the Earth has locked the Moon down so tightly, the Moon isn't able to affect, even through resonances, the Earth's obliquity. It is the other planets that affect the Earth's obliquity. But the other planets have little effect on the Moon because the Earth is so dominating.
