cross-purpose /(-p?r`p?s)/

cross·-pur·pose

cross-purpose

n.
  1. A counter or opposing purpose; a contrary aim; hence, that which is inconsistent or contradictory.
  2. A conversational game, in which questions and answers are made so as to involve ludicrous combinations of ideas.

Phrases & Compounds
To be at cross-purposes
to misunderstand or to act counter to one another without intending it; -- said of persons.