The most common and most likely answer to this crossword clue is the 13 letter word BACKSCRATCHER
I've seen this crossword clue in The Daily Telegraph
B
A
C
K
S
C
R
A
T
C
H
E
R
Buddy explains!
The straight part
We know that "one exchanging favours for mutual advantage" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one exchanging favours for mutual advantage = backscratcher.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BACKSCRATCHER mean?
BACKSCRATCHER noun- A long slender rod with a rake-like device on one end, sometimes in the form of a human hand, designed to let a person scratch their own back.