As it did in 2003 with Genet’s “The Blacks: A Clown Show,” Classical Theater of Harlem has once again tackled one of the modern canon’s gnarlier entries – and once again shown the neglected text to fine advantage. The lucky recipient this time: Adrienne Kennedy’s “Funnyhouse of a Negro.”

Sarah, the confused and wrathful young woman at the center of Ms. Kennedy’s harrowing 1964 psychodrama, has long, elegant fingers, beautiful eyes, and a pugnacious chin. Best of all, she informs the audience, her nose and lips are not overly Negroid.

There is, however, the matter of her hair. It’s “unmistakably Negro kinky hair.” And it’s falling out in clumps, as is Jesus’s hair. And Queen Victoria’s. And that of assassinated Congolese prime minister Patrice Lumumba. These notables coexist none too peacefully in Ms. Kennedy’s acid 1964 portrait of self-loathing and cultural dislocation – and they don’t intend to keep quiet.

Sarah (Suzette Azariah Gunn) may have a nightmarishly convoluted inner life, but director Billie Allen, who originated the central role more than 40 years ago, has created a lucid, bracing account of a proud, complicated woman capable of despising herself into the grave.

Ms. Allen consistently finds the logistical as well as the emotional logic in Ms. Kennedy’s impressionistic, at times impenetrable text. With its dreamlike tableaux (“Before him a bald head is dropped on a wire, someone screams”) and constantly looping language, “Funnyhouse” resists any sort of conventional through-line. Ms. Allen has created a plausible dreamscape that is absolutely hers, adhering to some of Ms. Kennedy’s conventions and ignoring others.

Large swaths of identical dialogue find their way into the mouths of different characters; Ms. Allen confronts this convention by having Sarah mouth the ricocheting phrases as they double and triple back on themselves, heightening their incantatory force. She gets an enormous amount of help from Kimberly Glennon’s superb costumes, which run the gamut from bloodied rags to resplendent gowns, not to mention the fistfuls of hair that fall from each character’s head. As with CTH’s “The Blacks” and “Ain’t Supposed To Die a Natural Death,” Ms. Glennon is as responsible for the company’s onstage magic as any actor or director.

Sarah is biracial and unable to consider herself white or black. This divided consciousness manifests itself in an onslaught of conflicting family narratives, many of which revolve around her father, who may have raped her white mother. “We are tied to him,” one of her personae explains, “unless, of course, he should die.” Consequently, depending on which version Sarah tells, he has been bludgeoned, hanged, shot, or crucified (or none of the above), and she is to blame for at least two of those deaths.

Lying morosely in her Upper West Side bedroom, Sarah longs for the genteel company of whites in order “to maintain a stark fortress against recognition of myself.” The fortress image proves prophetic: A series of sharp, menacing knocks are heard throughout. Is this the sound of Sarah’s father coming to either rescue or harm her? Of nails being driven into Jesus on the cross? Or of her darkest, most repressed thoughts, ones far more destabilizing even than the ones we see onstage?

That any of these scenarios remain equally viable is a testament to Ms. Gunn’s fearless performance. The emphatic use of her hands from the very beginning gives the performance precious little room to grow as the horrors mount (a rare lapse in direction on Ms. Allen’s part), but Ms. Gunn provides a tumultuous yet delicate center to the play’s surreal goings-on.

A play like “Funnyhouse” invites considerable experimentation on the director’s part. Some of Ms. Allen’s innovations come off well, like turning one romantic clinch into a scene out of “Gone With the Wind,” or having the two white characters offer squirm-inducingly square responses to a gospel-inflected sermon by Jesus himself. Others are arguably not worth the effort, as when Ms. Allen reduces Sarah’s white landlady (Alice Spivak) to a pandering, Rhoda Morgenstern-esque caricature.

Even these occasional gaffes do little to mute the bruising force of Ms. Kennedy’s play after more than 40 years. That insistent knocking we hear may be an era-defining play demanding to be heard again. Classical Theater of Harlem was wise to heed this demand; intrepid theatergoers would be wise to follow suit.

Until February 12 (645 St. Nicholas Avenue, near 141st Street, 212-868-4444).