From Longman Dictionary of Contemporary Englishcross-purposesˌcross-ˈpurposes noun → at cross-purposes
Examples from the Corpus
cross-purposes• At the moment we are sick of politics with one faction at cross-purposes with another.• The present system of managed floating does contain the possibility of direct conflict if official intervention is carried out at cross-purposes.• I wouldn't want us to find ourselves talking at cross-purposes. 