does the moon pull on the earth with one-sixth the force that the earth exerts on the moon?
No. Like the tension on a rope, there can be only one attractive gravitational force between two objects. Look at the equation: the output is only a single force: http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html
Also keep in mind Newton's Third Law: The force that A exerts on B is always equal in magnitude and opposite in direction to the force that B exerts on A.
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