House-sitter perhaps supposed valium stultified acid band's core line-up (10)

I believe the answer is:

politician

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'house-sitter perhaps' is the definition.
(eg someone who sits in the House of Commons)

I cannot understand how the remainder of the clue works.

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(Other definitions for politician that I've seen before include "Legislator" , "One in Westminster" , "US senator?" , "could be Congressman" , "Eg, an MP" .)

I've seen this clue in The Independent.

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