The best artists, without exception, hate being “understood.” They will fight to the death if they feel they are in danger of being explained away. Artist statements, though sometimes unavoidable, are anathema to them. (Why make art if you can put it in a statement?) Many sense that the best way to evade the obligation to “make oneself clear” is to create decoys, in the form of avatars, doubles or duplicates.

Marcel Duchamp excelled at this. Jasper Johns, though less theatrical, was his best student. What both realized is that “identity,” to the extent that it exists, always defeats description.

“The Double: Identity and Difference in Art Since 1900” at the National Gallery of Art is partly about just this. It also addresses double vision, copies, mirror reversals, shadows, twins and alter egos. That’s a lot to take on. But the show is concise, rigorous, funny and heartfelt. As such it’s an antidote to the noxious politics that today turn every word into a watchword against its opposite.

Better yet, it’s groaning with great art. Pablo Picasso, Giorgio de Chirico, Andy Warhol, Diane Arbus, Kerry James Marshall and Eva Hesse are just some of the artists included. There are surprises galore. (My favorite? A self-portrait, suggestively fractured in two, by Sylvia Plath. The poet made it at Smith College while writing her senior thesis on — what else? — the theme of “the double” in the novels of Dostoyevsky.)

Johns and his hero Duchamp underpin “The Double,” which was put together by James Meyer, the National Gallery’s curator of modern art. Both artists dismantled the notion that our identities are either stable or knowable. They immersed themselves instead in whirlpools of poetic secrecy, spirals of deflection, a circus of self-evasion.

The show is, as the subtitle rather flatly vows, about “identity and difference.” But don’t be put off. Meyer takes these cant words of our current era and all their unspoken implications (“you must express your assigned identity; you must celebrate difference”) to a deeper place. Transcending the infantilizing miasma of affinity groups, identity acronyms and the rote recitation of pronouns, the works in “The Double” take us to weirder, more provocative, more philosophical places.

Two artworks at the exhibition’s entrance appear to announce a political agenda. One is a double flag painting by Johns, the other a neon sign (the word “America” and its upside-down reversal) by Glenn Ligon. Johns has spent his career thinking through the implications of copies, pairs and doubles. Like targets and numbers, the flag was simply (as he put it) an image the “mind already knows.” He wasn’t trying to evoke a divided America. Foisting that reading on “Two Flags” may be tempting, but it’s glib.

Ligon’s “Double America,” on the other hand, clearly is political. It’s about America’s racial divide, and what W.E.B. Dubois referred to as the necessary “double consciousness” of African Americans. But Ligon is too subtle and grown-up to be making facile propaganda statements. Something deeper is going on in his work, and in the show generally.

National Gallery admission is free. Nonetheless, you make your way through the first galleries of “The Double” unable to shake off the constant, elated feeling that you are getting two for the price of one. After the Johns-Ligon prologue, we see two still lifes by Matisse. The first time he painted a motif, Matisse wanted to record his initial response; the second time, to distill and deepen it.

After the Matisses come two paintings of a chocolate grinder by Duchamp. One casts shadows, the other is flatter, more diagrammatic, with thread sewn into the canvas; a new proposition about the same thing. A little farther on are two versions of Arshile Gorky’s heartbreaking double portrait of himself with his late mother, followed by two near-identical abstractions by Robert Rauschenberg.

What is going on here? Why did these artists paint the same thing twice? And how do we know the copies aren’t fakes?

Rather than propositions about identity, the works in “The Double” are mostly expressions of curiosity. Some ask, at the most basic level, what it means to have two eyes instead of one, or what to make of the fact that our bodies are basically symmetrical — one side mirroring the other?

Others address technologies of reproduction which, with ever-increasing ease, turn one image into a copy of itself, a double. How, they ask, can our sense of ourselves as unique survive this rampant duplication? When a copy is made, is it the same as the original? Or does some quality (its “aura”?) escape? And what of love? Is love not also a manifestation of life’s inherent drive to duplicate?

By the end of the show, you find yourself wondering if all art isn’t an expression of the need to duplicate nature. That proposition is tackled head-on by René Magritte, whose 1933 painting, “The Human Condition,” shows a window with drawn curtains. In front of the window, exactly congruent with the landscape outside, is a painted landscape on an easel. “Every picture,” notes the wall label, “is a double of what it depicts.”

Much of modern art was an attempt to escape this truism — to make pictures that don’t depict anything and are therefore singular, unrepeatable. Hence abstraction. But Magritte suggests that art is always mimetic, if not of the external world then at the very least of consciousness.

One way to create a double — albeit in reverse — is to mirror the original. I was spellbound, in a section of the show about mirror reversals, by the Italian artist Alighiero Boetti. So obsessed with doubling was Boetti that he changed his name to Alighiero e Boetti (Alighiero and Boetti): no longer one artist but two. A two-minute video shows the artist writing on a wall with both arms simultaneously. The text inscribed by his left hand (“The Body Always Speaks in Silence”) mirrors and reverses that written by his right. Impressive feat!

Doubles create dilemmas: Of the two things before me, which do I prefer? A work by Felix Gonzalez-Torres, who used found objects (a la Duchamp) and modes of poetic minimalism to express aspects of same-sex love, comprises two stacks of white paper. The sheets in one stack are inscribed with “Nowhere better than this place”; those in the other with “Somewhere better than this place.” Visitors are invited to take a sheet with them — but which one?

Not far away is a tribute to Gonzalez-Torres and his partner, Ross Laycock, who died of AIDS-related causes in 1991. The work is by Roni Horn, who once stated her desire for “a language without pronouns” and approvingly compared the river Thames to “a solvent for identity.”

Gonzalez-Torres and Laycock had seen and loved an earlier work by Horn — a thin, crumpled sheet of annealed gold. So after Laycock’s death, Horn made a second work: this time two sheets of shimmering gold, one on top of the other. “There is sweat in between,” she told Gonzalez-Torres, who responded: “I knew that.”

In a context of AIDS and homophobia, even sweaty sheets are political. But if “The Double” is trying to teach us anything about politics, it may be that our current dysfunction is at least partly attributable to our preoccupation with crudely limiting “identities.”

The idea that to achieve justice people must unite and march under particular identity banners has led to incredible gains. But as these activist strategies have spread, they have tended to calcify, deepening divisions, inviting reaction, imperiling democracy. It may be that to extend justice today, and preserve democracy, we need to lower those banners and become more curious about one another.

And that, of course, is where art comes in. So it’s a relief that this show, in the heart of a city choking on politics, is above all about tenderness, humor, invention and love.

“The Double: Identity and Difference in Art Since 1900” at the National Gallery of Art through Oct. 31. nga.gov