Crossword Genius

Glad as bed is moved, revealing pouch by seat (9)

Ross

I believe the answer is:

saddlebag

I'm a little stuck... Click here to teach me more about this clue!

'pouch by seat' is the definition.
The definition and answer can be both man-made objects as well as being singular nouns.
Perhaps they are linked in a way I don't understand?

'glad as bed is moved' is the wordplay.
'is moved' indicates an anagram.
'glad'+'as'+'bed'='gladasbed'
'gladasbed' anagrammed gives 'SADDLEBAG'.

'revealing' is the link.

Can you help me to learn more?

(Other definitions for saddlebag that I've seen before include "Storage for riders" , "one lying over the wheel?" , "bike accessory" , "Something used to carry items when riding" , "storage place for bicycles" .)

I've seen this clue in The Times.
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