Reciprocal has both general and mathematical meanings. The general term describes an action done in response to, or as a returned favor of, an action by someone else. In math, reciprocal is the inverse of a number derived by dividing one by that number.
The expression, "You scratch my back, and I'll scratch yours," signifies an expectation of reciprocal behavior. The person making this statement conveys that if he or she does a favor for someone, he or she expects that person to return the favor in the future.
An example of a mathematical reciprocal is that 1/3 is the reciprocal of 3.