Why is the moon that special? Is it really a well built coincidence?
If the "perfect rotation" you describe is the fact that same face of the moon always faces the earth, this is explained by the 'Giant Impact Hypothesis' which also explains the formation of the moon. The molten earth was impacted by a large mass causing a part of the liquid earth to be flung out into space creating the moon. For a time, the two celestial bodies to be (the earth and moon) were connected to one another by some bridge of mass until eventually the centripetal force of their rotation finally pulled them apart. In fact, the two bodies are still moving apart today. 

https://en.wikipedia.org/wiki/Giant-impact_hypothesis