What we have in common, we share mutually. Some purists argue that mutual means something experienced or expressed by two people about each other, and so it is wrong to speak of something such as “our mutual enemy.” Their argument is that if you and I have an enemy in common, that adds up to a total of three people, and three is not two.
In the same manner, some believe it’s incorrect to refer to more than two people as making an agreement to their mutual benefit. The correct word, they say, is common (which I used as an example above).
However, not all of us are purists, and as word usage changes in society, those purists lose ground. Not many editors or grammarians are going to scream “violation!” if you use mutual for three or more people or things.
Now we often say things like “mutual understanding” and “mutual cooperation” when referring to groups of people.
Reciprocal can be used in place of mutual when it means given by each (of two)to the other, such as in reciprocal understanding.
More commonly, reciprocal is used to mean “given or done in return.” I do you a favor, you reciprocate in like manner. Reciprocal has the sense of equality, so that if you say a reciprocal action has been taken , it implies sameness or similarity. However, mutual action doesn’t necessarily mean the same action by two people; it means the people are acting in concert.
I hope we’re now in mutual understanding of all this.