Longman Dictionary of Contemporary English
cross-purposes
noun
EXAMPLES FROM CORPUS
▪ At the moment we are sick of politics with one faction at cross-purposes with another.
▪ I wouldn't want us to find ourselves talking at cross-purposes.
▪ The present system of managed floating does contain the possibility of direct conflict if official intervention is carried out at cross-purposes.
Wiktionary
cross-purposes
n. (cross-purpose English)
Usage examples of "cross-purposes".
He had always been decent enough with me, but then we'd never been at cross-purposes over anything.
Something about the possibility of us being at cross-purposes at some future time.
On the other hand, it was also critical that the American cavalrymen didn't work at cross-purposes with what the Cherokee warriors were going to be doing.
They dislike each other and work at cross-purposes but betrayal, corruption, any straying from whatever common path they have chosen, these are inconceivable.