From 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]