A year after Eddie Redmayne proved his incredible capacity for reinvention in “The Theory of Everything,” the freckle-faced Brit pulls off the ultimate identity overhaul as “The Danish Girl,” portraying gender-reassignment trailblazer Lili Elbe, nee Einar Wegener, who was one of the first to make a “sex change” via surgery. For an actor, there can be few more enticing — or challenging — roles than this, in which the nature of identity, performance and transformation are all wrapped up in the very fabric of the character itself, and Redmayne gives the greatest performance of his career so far, infinitely more intimate — and far less technical — than the already stunning turn as Stephen Hawking that so recently won him the Oscar. Reuniting with “Les Miserables” director Tom Hooper in a return to the handsome, mostly interior style of the helmer’s Oscar-winning “The King’s Speech,” Redmayne finds himself at the heart — one shared by Alicia Vikander, as Einar’s wife, Gerda — of what’s destined to be the year’s most talked-about arthouse phenomenon.

Though set nearly a century ago, between the years 1926 and 1931, it has taken this long for the subject to receive such a high-profile treatment, and though some might say argue it comes as too little too late, the pic’s release could hardly be timelier in the wake of so many recent headlines — especially the legalization of gay marriage and Caitlyn Jenner’s high-profile gender transition. As it happens, “The Danish Girl” has been in the works since the publication of David Ebershoff’s novel 15 years ago, with Nicole Kidman originally attached to play Lili for director Lasse Hallstrom.

Clearly, this was never not going to be a “prestige” picture. And while that ultra-respectful approach will engender allergic reactions in some, who’d sooner see a gritty, realistic portrayal — a la Jill Soloway’s terrific “Transparent” series for Amazon — than one seemingly tailored for the pages of fashion and interior-design magazines, there’s no denying that Hooper and screenwriter Lucinda Coxon have delivered a cinematic landmark, one whose classical style all but disguises how controversial its subject matter still remains. For rowdier crowds, there will always be “Myra Breckinridge.” In order to penetrate the conversation of “polite” society, however, one must play by its rules, and “The Danish Girl” is nothing if not sensitive to how old-fashioned viewers (and voters) might respond, scrubbing the story of its pricklier details and upholding the long-standing LGBT-movie tradition of tragically killing off the “monster” in the end.

Pause for a moment to consider how significant a choice it was to cast a man, Redmayne, in the lead role — which is not to say that gifted actresses haven’t delivered incredible work in pre- and post-op male-to-female parts, from Felicity Huffman’s “Transamerica” road-tripper to Olympia Dukakis’ “Tales of the City” landlady. But it’s almost unfair to cast according to the character’s target gender, as it inoculates whatever resistance the public feels toward these procedures (although one day, Hollywood will cross the hurdle of inviting trans actors to play such roles, as well as those on either side of gender divide).

When “The Danish Girl” first introduces Redmayne’s character, he is dapperly costumed as a Danish gentleman, making eyes across a gallery opening at his wife, Gerda (Vikander). And what eyes! Throughout the actor’s career, casting directors have always wrestled with Redmayne’s exceptionally specific look, which is not so much androgynous as a paradoxical blend of pretty and off-putting features: those unblinking, long-lashed eyes; the sharp, knobby cleft of his nose; elegant malar bones set above pale, sucked-in cheeks; and lips to make Angelina Jolie jealous. Our brains never quite know how to process Redmayne’s appearance, and here, Hooper takes full advantage of that situation.

The first time Einar dons ladies’ clothes, the idea is Gerda’s: Already married, the couple both make their living as artists, and though Einar’s work is taken seriously, a gallerist tells Gerda that she could be great, if only she found the right subject matter. It’s just an offhand suggestion, a favor really, but while waiting for her model to arrive, Gerda asks her husband to slip on a pair of ivory stockings and matching silk pumps, inadvertently releasing her muse.

When dancer friend Ulla (Amber Heard) does appear, she responds to the situation with delight, christening Einar’s alter ego “Lili.” It’s a confusing moment for Einar, who has long repressed what made him different from the other boys in Vejle, Denmark, and who will later tell his wife, “You helped bring Lili to life, but she was always there.” Outsiders always want to know what makes LGBT people “that way,” seeking psychological answers to a situation with which they can’t identify, but “The Danish Girl” dutifully avoids any such armchair diagnosis. It is surely for the benefit of such skeptics that Lili explains, “God made me this way, but the doctor is curing me of the sickness that was my disguise.”

Until scandalously recent times, the medical community’s response to such identity issues was to diagnose their “aberrant” or “perverse” patients as schizophrenic or insane and to shock, drug or irradiate the sickness out of them. That chronic misunderstanding becomes a running thread in the film, which tends to be far more pleasant to watch when Lili is getting to be herself. She is understandably hesitant to emerge at first, though Einar (who hates public gatherings) agrees to accompany his wife dressed as Lili, his imaginary ginger-haired cousin from Vejle. The resulting scene may be the film’s best, a coming out as thrilling as Cinderella’s ball, in which Lili can feel the gaze of everyone in the room on her.

This, he learns, is how beautiful women feel all the time in public, and if audiences take nothing else away from Hooper’s humanely empathetic film, this lone gender-swapping lesson in identification is victory enough. Naturally, Lili’s situation is more complicated, instantly escalating when a young suitor named Henrik (Ben Whishaw, masculinized for contrast’s sake) takes Lili aside and tries to kiss her — at precisely the moment Gerda comes to fetch her husband. Clinging to the notion that Einar and Gerda’s love was strong enough to weather all the challenges of his transition, Coxon’s screenplay is dramatized in such a way that the couple never discuss any of these setbacks immediately, but typically get around to it a scene or two later, back at home with their pet dog to distract and dressed in a fresh set of costume designer Paco Delgado’s lovely frocks.

In this case, Lili has vanished by the next morning, replaced by Einar, who appears to be genuinely wrestling between the two personae struggling for control of his body. At one point, reunited with boyhood friend Hans (Matthias Schoenaerts), he admits that he has considered suicide, but held back because he understood he would be killing Lili at the same time — a sentiment that all too many trans people share and one of the many reasons such a well-rounded portrayal is long overdue.

Of course, Einar’s struggle is very real, all the more difficult for its time, given the prevailing homophobia (dramatized in a Parisian gay-bashing) and sexual politics of the time. The late ’20s were still early days for women’s rights, and Redmayne represents someone trying to follow his female intuition at a time when that meant ceding the social privileges of manhood — an irony made clear in Gerda’s character, whose own bisexual identity has been conspicuously omitted, so as not to complicate the film’s politics.

Spotlighting the least-represented thread in the LGBT quilt, “The Danish Girl” clearly wants to untangle the trans experience from the blanket definition of homosexuality, using Lili’s rejection of Whishaw’s gay character and her interview with gender-confirmation surgeon Dr. Kurt Warnekros (Sebastian Koch, playing the sensitive pioneer) to distinguish the two. What’s of utmost importance here is the discovery and ultimate acceptance of Lili’s true identity, and from the film’s perspective, the gender question has nothing (or very little) to do with sex. Rather, it’s something that reveals itself at first in mirrors and other reflective surfaces, and later directly to camera, as Redmayne explores Einar’s hidden second persona.

As Hans puts it at a train sendoff that recalls “Casablanca,” “I’ve only really liked a handful of people in my life, and you’ve been two of them.” But Lili’s emergence is a gradual and hesitant process, beautifully embodied by Redmayne — and reflected by Vikander, whose Gerda does her best to adapt alongside her husband, amounting to a substantive role for the film’s resident “Swedish girl.” Shy at first, like a flower opening, Redmayne ducks his eyes and turns his head as Lili, his confidence growing in tandem with the rolling boil of Alexandre Desplat’s strings and piano score.

Though his first attempt at makeup looks rather grotesque, he becomes quite the pro (with an assist from actual pro Jan Sewell, who also designed the star’s prosthetics in “The Theory of Everything”), upstaging the other women whenever he goes out in public. At first, the goal is simply to pass — a game, almost — but in time, the butterfly motif becomes clear, reflected in the pic’s ripening color scheme. By the end, the goal is complete and total transformation as Einar studies the body language of the women around him and incorporates them into what for Redmayne is a character, but for Lili is her true self.