The Collaborative International Dictionary of English v.0.48:

cross-purpose \cross"-pur`pose\ (-p?r`p?s), n.
   1. A counter or opposing purpose; a contrary aim; hence, that
      which is inconsistent or contradictory. --Shaftesbury.
      [1913 Webster]

   2. pl. A conversational game, in which questions and answers
      are made so as to involve ludicrous combinations of ideas.
      --Pepys.
      [1913 Webster]

   To be at cross-purposes, to misunderstand or to act counter
      to one another without intending it; -- said of persons.
      [1913 Webster]

WordNet (r) 3.0 (2006):

cross-purpose
    n 1: a contrary aim; "at cross-purposes"