cross-purpose
cross-purpose (-p?r`p?s) , noun
1.
A counter or opposing purpose; a contrary aim; hence, that which is inconsistent or contradictory. — Shaftesbury
2.
A conversational game, in which questions and answers are made so as to involve ludicrous combinations of ideas. — Pepys
Collocations (1)
To be at cross-purposes , to misunderstand or to act counter to one another without intending it; -- said of persons.