Band brought back "Bits & Pieces" (5)

I believe the answer is:

strap

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'band' is the definition.
(strap is a kind of band)

'brought back bits pieces' is the wordplay.
'brought' becomes 'arts' (I am not sure about this - if you are sure you should believe this answer much more).
'back' is a reversal indicator.
'bits' suggests taking the first letters (I've seen 'bit of' mean this).
The first letter of 'pieces' is 'p'.
'arts' reversed gives 'stra'.
'stra'+'p'='STRAP'

Can you help me to learn more?

(Other definitions for strap that I've seen before include "Belt; leather strip" , "e.g. Belt" , "Leather thong" , "Whip" , "loopy band" .)

I've seen this clue in The Independent.

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