A division secretly devising gadgets to clean up Muslim dress (5)
I believe the answer is:
burqa
'a division secretly devising gadgets' is the definition.
I know nothing about this answer so I cannot judge whether this works.
'clean up muslim dress' is the wordplay.
'clean' becomes 'rub' (as in rubbing something to clean it).
'up' shows that the letters should be reversed in order (in down clue: letters go upwards).
'muslim dress' becomes 'qa' (I am not sure about this - if you are sure you should believe this answer much more).
'rub' reversed gives 'bur'.
'bur'+'qa'='BURQA'
'to' is the link.
Can you help me to learn more?
(Another definition for burqa that I've seen is " religious garb".)