cross-purpose n.

word-forming element typically representing cross as a noun, adverb (cross-examine), adjective (crossbar), and in many words a confluence of them. "There is no distinct line of division between cross as an adjective and cross as a prefix. As a prefix, it often represents the adv. cross, or the prep. cross, across." [Century Dictionary]

c. 1300, purpus, "intention, aim, goal; object to be kept in view; proper function for which something exists," from Anglo-French purpos, Old French porpos "an aim, intention" (12c.), from porposer "to put forth," from por- "forth" (from a variant of Latin pro- "forth;" see pur-) + Old French poser "to put, place" (see pose (v.1)).

Etymologically it is equivalent to Latin propositium "a thing proposed or intended," but evidently formed in French from the same elements. From mid-14c. as "theme of a discourse, subject matter of a narrative (as opposed to digressions), hence to the purpose "appropriate" (late 14c.). On purpose "by design, intentionally" is attested from 1580s; earlier of purpose (early 15c.).