It does. The Earth and Moon both rotate "around each other" as modelled here (from Wikipedia):
Note that this is just and example model. The effect on Earth is in reality even smaller than visualized here (and the Moon's orbit is not perfectly circular).
Had the two objects had the same masses, then the orbital motions would have been equal:
The issue is what "rotating around something" exactly means. You could say that neither of the objects rotate around each other in any of the cases shown - because in fact they rotate around their shared "mid-point", so to say. This point is called the barycenter. The barycenter is the point that their gravitational influences "average down to", if we were to imagine a stationary non-orbiting object that they both rotated around.
A smaller (less massive) object gives a weaker gravitational pull, thus causing a smaller centripetal acceleration of the more massive object, giving it a smaller orbit and smaller orbital speed. This is the case for the Earth-Moon system.
Although the mechanism is the same, and they both still rotate around the barycenter, the more massive Earth is rather "wobbling" than rotating/orbiting. It is still orbiting about the barycenter, but that barycenter is located inside it not far from it's own centre.
Some good illustrations of the real barycenter location in the Earth-Moon system are found in this answer on the Astronomy SE site, visualizing how close that barycentre is to Earth's own centre and therefore how little an effect the Moon has on Earth.