The phrase "a mutual exchange" is correct and usable in written English.
It can be used to describe a situation where two parties give and receive something of value to each other.
Example: "The agreement was based on a mutual exchange of resources that benefited both companies."
Alternatives: "a reciprocal exchange" or "a two-way exchange".
I manage a service that helps people move by swapping homes via a mutual exchange.
As part of a mutual exchange, a Japanese ship will visit China later.
It didn't disappoint: marvellously acute portraits that establish a mutual exchange between sitter and viewer half a century before Velázquez.
Nothing should be stranger in our way of living and allowing for a mutual exchange of ideas between our communities, than a model of parallel lives, shielded beneath an illusion, which in reality is of mutual ignorance.
Leary was one of the first people to have a site on the world wide web, and that model of a mutual exchange of information is very much with us today.
He embarked on a mutual exchange of insults with Mr Gotovina's lawyer on twitter and has proposed a General Assembly debate on ad hoc UN war crimes tribunals next year, by a strange coincidence on the anniversary of the founding of the Croatian quisling state in 1941.
I have discovered that respecting the energy of others and ourselves is important when it comes to any mutual exchange of energy.
We see him and he sees us: a startlingly mutual exchange.
This example also shows that the Americanisation of West German society after 1945 was a process of mutual exchange in which care packages played an important role.
How you notice the other person, how you acknowledge the other person, how the other person notices you and acknowledges you... all constitute a kind of mutual exchange.
The development of interventions has borrowed from the intervention mapping approach [ 19] and has taken place through a process of mutual exchange of information and experiences among all partner institutions.
