Hi Craig,
Understand HCM operates as sub modules like OM , PA , PY , TIME.
Your Cost Centers fall under OM side of HCM.
OM work with Object Types with some identifiers and relationships.
Your Cost Center is K and relationship is with Org Unit (O) or Position (S)
Org Unit to a Cost Center is related using A011 Relationship through IT1001.
Cost center assignments are inherited along the organizational structure.
With principle of Inheritance if you have assigned a O-K relationship at the Org Unit Level, all the reporting Org Units & Positions will inherit the Same Parent Cost Center
In cases of O-K / S-K Relationship is changed for any other org Units or Positions... in such cases, the reporting org units or positions will inherit.
Check these Tcodes :
PA20 - Any PERNR - Infotype - 0001 - Org Assignement - check Enterprise Structure block in that screen.
PP01 - Take Object - Cost Center or Org Unit or Position - check Relationships .
Hope it helps.
Regards
Rohit.S