Strangeness of headless corpse Inspector West finally cracks (6)
I believe the answer is:
oddity
'strangeness' is the definition.
(I know that oddity is a type of eccentricity)
'headless corpse inspector west finally cracks' is the wordplay.
'headless' means to remove the first letter.
'corpse' becomes 'body' (I've seen this before).
'inspector' becomes 'di' (detective inspector).
'finally' indicates one should take the final letters.
'cracks' is an insertion indicator (some letters create an opening in the word).
The final letter of 'west' is 't'.
'body' with its initial letter taken off is 'ody'.
'di'+'t'='dit'
'ody' going around 'dit' is 'ODDITY'.
'of' is the link.
(Other definitions for oddity that I've seen before include "Toy did (anag.)" , "Peculiar thing or person" , "Something strange or eccentric" , "Quirk" , "Something strange or peculiar" .)