Hands over exchanged gift (4)
I believe the answer is:
swap
'exchanged gift' is the definition.
(I've seen this before)
'hands over' is the wordplay.
'hands' becomes 'paws' ('paw' can be a synonym of 'hand').
'over' is a reversal indicator.
'paws' back-to-front is 'SWAP'.
(Other definitions for swap that I've seen before include "Fifty-fifty odds" , "Exchange; hands back" , "Exchange for paws" , "Switch" , "Change over" .)