Holds things together after play, oddly, recalling agent suggesting cut (9)

I believe the answer is:

paperclip

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'holds things together' is the definition.
I can't judge whether this definition defines the answer.

'play oddly recalling agent suggesting cut' is the wordplay.
'oddly' indicates alternate letters (letters in odd-numbered positions).
'recalling' says the letters should be written backwards.
'agent' becomes 'rep' (short for representative).
'suggesting cut' becomes 'clip' (I am not sure about this - if you are sure you should believe this answer much more).
The alternate letters of 'play' are 'pa'.
'rep' reversed gives 'per'.
'pa'+'per'+'clip'='PAPERCLIP'

'after' acts as a link.

Can you help me to learn more?

(Other definitions for paperclip that I've seen before include "Document fastener" , "Fastener for documents etc" , "Small piece of office equipment" .)

I've seen this clue in The Irish Times.

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