In many Oceanic languages the "indirect" possessive construction, which is typically associated with alienable possession, uses special forms to host person and number agreement indexing the possessor. This can be contrasted with the "direct" possessive construction, typically associated with inalienable possession, where a lexical possessum noun itself carries possessor-indexing agreement. The host forms used in the indirect construction are often referred to as "classifiers." We argue that this term should not be applied to indirect possession marking in many Oceanic languages, and present evidence to show that indirect possessor-indexing hosts in such languages do not have the properties typically associated with classifiers. In contrast with this, we further argue that these indirect possessor-indexing hosts should be treated as the syntactic head of the noun phrase in which they occur, thereby allowing treatment of the syntax of NPs with indirect possession that is consistent with those with direct marking. In both instances, the person and number indexing morphology simply attaches to the syntactic head.