For a period while living in Chile with her artist-husband Lee Mullican in the late 1960s, Luchita Hurtado painted inside a walk-in closet, standing there and looking down over her breasts and belly to her feet and the floor below. Sometimes a bar of light came in through the slats of the door. She included this in her painting, too. She looks down at the rug, or into a woven basket, that streak of light picking out the weave of the basket, momentarily brightening the pattern on the Navajo rug.

In one painting she drops a strawberry into a bowl on the floor. It hangs in mid-air in the half-lit gloom. Sometimes there seem to be two, three or even four people in there: eight feet on a gorgeous rug, green apples big as bowling balls, inexplicably huge and vivid against the dyed lozenges of the rug, every nub in the rug’s structure picked-out in paint. The painting evokes the feel of the rough texture against naked feet. In another it is all flat zigzags and pattern, interrupted by naked mellow skin.

Facebook Twitter Pinterest Untitled, 1971. Photograph: © Luchita Hurtado/courtesy the artist and Hauser & Wirth

Hurtado’s self-portraits without a mirror, painted in tight spaces, have an introspective, surprising intimacy. Imagine a woman looking down, at her own body and beyond herself, even when there is nothing there except the floor beneath her feet. You never see her head. Looking down, she views herself. This is but one strand of Hurtado’s varied, contradictory, startling career.

At this period of her long and fascinating life, Hurtado had nowhere else to be alone to paint. Here’s her hands, one holding a cigarette, the other a flaring match. She painted overhead views of hands holding drinks, in a naked convivial gathering, diagrammatic couplings, knees and elbows, breasts and torsos as a female landscape of folds and angular hillocks. In one troubling drawing, a group of men in suits and ties loom over her, as though coming into view over the horizon of her torso under a penumbrous grey sky. When she does confront her own face, she paints it as other and imperious.

This exhibition is the Venezuela-born artist’s first solo show in a public gallery. She is 98, still working, smart and funny and full of stories, angry about the state of the world but looking forward not back.

What a life Hurtado has had. Born outside Caracas, Hurtado was brought to to New York when she was eight by her mother. She never saw her father again. Married at 18, she moved to Santo Domingo, having later to cross the Dominican Republic’s border to Haiti to escape the advances of Héctor Trujillo, head of the Dominican army, and thence back to New York. This is but one arc of her peripatetic life, which has seen her living in New York and California, Mexico, New Mexico and Rome. If you ask her where she belongs, she says she is a terrestrial.

Facebook Twitter Pinterest Luchita Hurtado in her Santa Monica home and studio. Photograph: Oresti Tsonopoulos/© Luchita Hurtado/courtesy the artist and Hauser & Wirth

Along the way, she befriended the Japanese-American artist Isamu Noguchi, hung out with Buckminster Fuller and Willem and Elaine de Kooning. The roll-call of her friendships and encounters with American and exiled European artists and writers, oddballs and intellectuals, is extraordinary. Chagall, Léger, Miró, André Breton, Frida Kahlo and Diego Rivera, Jane Bowles (author of the wonderful Two Serious Ladies), Rufino Tamayo.

It goes on. Surrealist Leonora Carrington built a house from cardboard boxes for Hurtado’s children to play in. Marcel Duchamp massaged her feet. She found Jackson Pollock alarming. Her second husband, Wolfgang Paalen, wrote a play, and Mark Rothko, Robert Motherwell and Lee Krasner all took part. Her second son died from polio. With her third husband, painter and teacher Lee Mullican, she gave birth to Matt Mullican, who became an artist, and his Los Angeles-based film-writer and director brother John. It appears that Lee Millican, a painter of tessellated, complex all-over paintings, took up most of the physical and mental space.

The constraints of marriage, motherhood and the emotional and reproductive labour of the artist’s wife never stopped her working, though the difficulties were clear. For many years she rarely showed her work to anyone. Encouraged by other female artists, including Joyce Kozloff and Judy Chicago, she did exhibit sporadically in the 1970s and 80s, and in the 80s the Guerrilla Girls asked her to join them. But she never was a joiner, I think. Hurtado’s late discovery, or rather rediscovery, was pure chance. Ryan Good, director of Lee Mullican’s estate, came across many of her paintings, initialled LH, stacked among Lee’s work. Who is LH, he asked? That’s me, she said.

Facebook Twitter Pinterest The Umbilical Cord of the Earth is the Moon, 1977. Photograph: Jeff McLane/© Luchita Hurtado/courtesy the artist and Hauser & Wirth

Hurtado’s Serpentine show includes perhaps a tenth of her output, but follows its main themes from the 1930s to the present. The earliest work is one of the most compelling: the burner of a gas hob, the pale blue flame a halo beneath the cast-iron pan-support, seen from above. Drawn in graphite and coloured pencil, it is as plain and modest as its subject. The blue glows faint and with a quiet energy in the centre of the image. Drawn in 1938, when Hurtado was in her late teens, it looks forward, circuitously, to both the paintings of her body and to a number of works from the 1970s – white on bare cotton and linen, all of which have a direct symmetry and attack. The words WOMB and EVE, WOMAN and YOU are repeated. Other word-paintings fizzle into an alphabetical abstraction of repeated lines and curves, a breakdown of language into highly compressed, multicoloured vectors.

Vitality, tenderness, spookiness, intimacy, gawkiness, sexiness, subtlety, anger, jazzy abstractions, totemic figures, near monochromes, word paintings and the acutely observed come one after the other. A silhouetted figure fogs a window with their breath, or blows smoke. The exhaled cloud is like a skull. Above it hangs a trompe l’oeil feather, a symbol of good luck. More feathers drift in a Magritte-like sky.

In a recent series, Hurtado looks across her supine body, a head emerging between her raised legs. In the best of these, she’s as rough and cursory as Marlene Dumas. Word paintings return, as slogans. She keeps going. She persists, indomitable.

• Luchina Hurtado: I Live I Die I Will Be Reborn is at the Serpentine Sackler Gallery, London, 23 May-20 October.