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Forget Athens and Sparta, Sodom and Gomorrah, Dallas and Fort Worth. Of all the cities whose destinies have been twinned for one reason or another, few can match the intensity of the love-hate relationship between Paris and New York. For much of the past century, the City of Light and the City That Never Sleeps have competed feverishly for leadership in culture, couture and coolness, even as they freely exchanged influences and expatriates.

Nowhere has that fond rivalry been more evident than in the field of design. "Paris/New York," a dense, imaginative exhibit running at the Museum of the City of New York until Feb. 22, focuses on 1925-40 — the bright, anodized moment when Paris and New York were forging new ways of looking good. That was when Paris invented Art Deco (and New York improved on it), New York was alive with a new sound called jazz (and Paris went crazy over it) and Paris dominated haute couture (while New York industrialized it). "Let's work together," enjoined the French architect Le Corbusier. "Let's build a bridge across the Atlantic." Leading artists and designers of the day took up the challenge, hopping back and forth across the pond to create paintings, posters, buildings, furniture, fashions, dinnerware, interiors, jewelry and luxury liners.

Aside from a few paintings by the transatlantic practitioners of Neo-Romanticism, a gentler version of Surrealism, this is a show about stuff. One of the first things you see is a 6-ft.-long (2 m) wooden model of the Normandie, that floating showcase for Art Deco and French luxury that was once the classiest way to go between the two cities. Nearby are modernistic silver serving pieces and other shipboard relics. A striking 1934 photomontage advertising the Normandie shows it sailing through Times Square past the Art Deco Paramount Building. Art Deco — that decorative fusion of Art Nouveau, Constructivism, Cubism, Modernism and Futurism — made its debut at Paris' 1925 International Exposition. In a New York minute, the style took Manhattan and was replicated in interiors, fabrics, typefaces and that local speciality, the skyscraper.

Various 1920s and '30s New York high-rises are represented in photos, ironwork and hunks of decoration — especially Rockefeller Center, the 22-acre (nine hectare) living museum of Art Deco that lies 52 blocks due south of the exhibit. The French government, not coincidentally, was one of the center's first tenants. Indeed, France fell in love with the skyscraper, and the show includes plans for (mercifully unbuilt) Parisian versions that somehow lacked the energy of their New York counterparts.

Energy was America's competitive advantage. While Paris was bursting with ideas, New York was getting them built. The Paris furniture and fashion designers who were setting world standards began to find their masterpieces knocked off by New York department stores and the emerging Seventh Avenue garment district. A dressing table by French designer Léon Jallot is a riot of color and molded wood — oops, that's the copy made in New York for Lord & Taylor's department store.

If architecture is frozen music, as Goethe aphorized, then jazz is Art Deco on ice. France adored that American musical invention and especially Josephine Baker, the black American singer and dancer who electrified the Paris jazz scene in the 1920s. Her sleek, exotic beauty is on display in Art Deco – influenced posters, paintings, advertisements and fabrics, plus a film loop of her doing an athletic shimmy.

"There's a lot of text here," says "Paris/New York" curator Donald Albrecht of his show's detail-crammed signs and labels. "It's like a magazine article exhibit-ized." And, like any good magazine article, it has not just a beginning and middle, but also an end. One of the last things you see is a 1932 replica of a never-built luxury liner by Norman Bel Geddes, who — along with the likes of Gilbert Rohde and Donald Deskey — formed a rising group of distinctively American designers. Bel Geddes' model is the same size as the Normandie at the show's entrance, but sleeker and more futuristic — a metaphor for the passing of cultural leadership to New York.

Where it mostly remains. World War II ended Paris' heyday as the world capital of design, and indeed of the arts generally. Despite a brief renaissance in painting and film in the 1960s, Paris is today a beautiful city with great museums, a glorious past and a lot of visiting New Yorkers.

They should take note. The next culture-and-design rivalry will probably involve Beijing and Shanghai. Masterpieces by American creators of fashion, furniture and consumer electronics are already being replicated in the Yangtze and Pearl River deltas and sold at big-box stores. Meanwhile, Chinese painters like Yue Minjun and Zeng Fangzhi are pulling in big bucks at auction, and Chinese filmmakers, writers and musicians are not far behind.

Perhaps appropriately, the Normandie ended its days in New York. While being refitted as a troop ship in 1942, it caught fire, capsized and sank in the Hudson River at 47th Street. At war's end it was sold for scrap. But along with other reminders that the two great cities were once joined at the hip of all that was hip, the Normandie lives on in wood, silver and memory. Rivalries end, style endures.

• Don Morrison