Catch up and split up (4)
I believe the answer is:
part
'split up' is the definition.
('part' can be a synonym of 'split up')
'catch up' is the wordplay.
'catch' becomes 'trap' (trapping is a kind of catching).
'up' says the letters should be written in reverse (in down clue: letters go upwards).
'trap' backwards is 'PART'.
'and' is the link.
(Other definitions for part that I've seen before include "Portion, segment" , "Section of a whole" , "Member" , "Portion of a whole" , "Component; divide" .)