With aims or goals that conflict or interfere with one another, as in I'm afraid the two departments are working at cross purposes. This idiom, first recorded in 1688, may have begun as a 17th-century parlor game called “cross-purposes,” in which a series of subjects (or questions) were divided from their explanations (or answers) and distributed around the room. Players then created absurdities by combining a subject taken from one person with an explanation taken from another.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.