Martin McDonagh has turned his attention to the swinging ’60s, but don’t expect Twiggy to show up in a miniskirt.

It’s a very different kind of swinging McDonagh has in mind, the kind that takes place at the end of a rope. “Hangmen,” his new play at the Off-Broadway Atlantic Theater Company’s Linda Gross Theater, is set on the day capital punishment was abolished in Britain.

Clearly, McDonagh is having a moment. His movie “Three Billboards Outside Ebbing, Missouri,” swept the Golden Globes and got seven Oscar nominations. “Hangmen” was all but a sellout before the first preview, leading to speculation (as yet unconfirmed) of a move to Broadway.

Soon as they can find a theater, would be my guess. The dark comedy (in the McDonagh tradition) is an entertaining, gripping mystery at heart, full of the kind of twists that, were it a novel, would keep you up turning pages all night.

Mark Addy, as formidable here as when playing King Robert in “Game of Thrones,” is Harry, known as the second-best hangman in Britain. In a short prologue, he presides with haughty bluster over the execution of a hapless soul called Hennessy (Gilles Geary, wringing a lot of pathos and some laughs out of his few minutes onstage), who proclaims his innocence until he no longer can.

Fast forward two years to Harry’s pub, where a reporter along with a gaggle of regulars are keen on hearing what Harry has to say about the demise of his grim occupation. A stranger (Johnny Flynn, reeking malevolence) comes in, obviously out of place, setting the plot into overdrive as he flirts first with Harry’s gin-swilling wife (a fine Sally Rogers), then his 15-year-old daughter (Gaby French, perfectly capturing teenage angst) who soon thereafter goes missing. Any more would spoil too much.

So let’s talk about the historical accuracy of this play, tautly directed by Matthew Dunster, who also staged the successful run at the Royal Court in London. While not exactly ripped from the headlines, it is based on fact. Names have been changed, except that of Albert Pierrepoint (Maxwell Caulfield), the No. 1 hangman, whose 11th-hour appearance brings the play to its not-so-surprising conclusion.

Is there an underlying indictment of capital punishment here? “It’s the courts that’s hanging ya, not us,” Harry tells the condemned in the prologue. But the play leads us to believe Hennessy might well have been an innocent man, leaving the audience to answer that one on its own.