Brendan Gleeson, Don Cheadle, Liam Cunningham, David Wilmot, Rory Keenan, Mark Strong, Fionnula Flanagan, Dominique McElligott, Sarah Greene, Katarina Cas (Written and directed by John Michael McDonagh; Sony Pictures Classics)

They are an odd, going-on-bizarre couple. One, played by Brendan Gleeson, is a sloppy, alcoholic, obscenity-spewing, hooker-cuddling, politically incorrect Irish policeman; the other, played by Don Cheadle, is a dapper, cerebral, eloquent, self-disciplined—if occasionally clueless--African-American F.B.I. agent. Together, yet seldom in sync, they set out to track down and defuse three explosive, torture-happy villains who have smuggled a half-billion dollars worth of drugs into Ireland and gleefully murdered a sensitive rookie cop in the process.

Can this impossible team of two survive its bloody mission and eventually build a beautiful friendship, and will you really care one way or the other? Depending on your tolerance for lewd language, graphic violence, slapstick sex, Irish shtick and brazen cinematic mischief, the answer to both questions is yes. Wait, better make that maybe.

You’ll know what I mean by maybe if you decide to take a chance on this constantly surprising, darkly comic, ultimately tender thriller. Working from his own screenplay, John Michael McDonagh makes a remarkable directorial debut, nimbly rotating and sometimes mixing elements of terror and humor, valor and grief.

And when it came to casting his first feature, McDonagh certainly had beginner’s luck. From bottom to top, from villains to heroes, each performer hits the precise right note. Playing the foul-mouthed, possibly soft-hearted Irish cop, Brendan Gleeson is sensational, perhaps even topping his “In Bruges” turn, which was written and directed by Martin McDonagh, brother of John Michael. As always, the splendid, mercurial Don Cheadle makes a deep impression as Gleeson’s (mostly) calm and reasonable partner in crime-fighting.

A very special bonus: The incandescent Fionnula Flanagan plays Gleeson’s vulnerable yet valiant, Irish-to-the-core old mum. And like her wayward son, she cherishes nothing more than a nip now and then.