Halfway through this magnificent exhibition hangs a portrait by Goya so famous it has its own nickname – The Black Duchess. It shows a fiercely intelligent woman standing outdoors in a romantic landscape dressed in crackling black lace. Her eyes flash, her sash blazes scarlet, the yellow and gold of her bodice burn like flames through the lace. The Duchess of Alba is as beautiful as she is haughty. One hand on hip, she points down with the other to some words written in the sand beneath her feet. Solo Goya – Only Goya. Is it a pledge of love, or an imperious summons, as if the artist should kneel before her? He puts himself, and us, on eye level with her gilded slippers. The duchess was his patron and friend, and perhaps more. What the portrait – and its subject – meant to him may remain forever undisclosed but Goya never let the painting go. It was with him until his death. This masterpiece is seldom seen except by habitués of the Hispanic Society Museum & Library on 155th Street and Broadway in Upper Manhattan. Founded in 1904 by the wealthy philanthropist Archer M Huntington, this is the most expansive collection of Iberian culture anywhere outside Spain. Anyone who has walked through the dark-wood galleries knows the sudden elation of coming upon El Greco, Zurbarán, Velázquez, Goya and more, but now the experience is available at the Royal Academy, while the Hispanic Society closes its doors for restoration. A whole museum, in essence, has been transported to Britain. Silver bracelets that once spiralled up the arms of Celtiberian women centuries before the birth of Christ gleam in the spotlit darkness. The face of Pan, horns curving sinuously from his shaggy bronze locks, stares out of a lamp found in a Roman villa in Málaga. A belt buckle made of garnets and glittering green glass throws up the sensation of the man (or woman?) who wore it, this great heavy work of art, strung across their navel. The people of the past are everywhere apparent. Huntington collected across four millennia and in every medium. He bought Giovanni Vespucci’s elaborate nautical map of the world, made in Seville in 1526, with its great parchment-white oceans and curious illuminated details, such as the harvest of nut wood along the coast of Brazil. He acquired medieval door knockers featuring crab claws, dragons and the heads of bats; Alhambra silks shining with eight-pointed stars and gleaming Valencian lustreware. A deep plate, made in Manises in the 1370s, its cobalt and gold patterns shimmering like nacreous shell, stretches across nearly half a metre. Huntington learned Arabic to understand Spain’s Moorish past, and studied the country’s colonial history. Here are Mexican portraits and manuscripts showing the devastating encounters between Indigenous Americans and invading conquistadors. Bowls made of black micaceous clay in Tonalá, Mexico, are sculpted inside with all kinds of fish, fronds and serpents. To drink from the waters of the vessel was to come upon an undulating subaqueous world. Everything here is so unexpected. You think you are looking at a tiny oil portrait on a bit of card that resembles nothing so much as an El Greco only to discover that’s exactly what it is – the rarest of miniatures. A Last Supper from Bolivia, meanwhile, is worked in oil paint and inlaid mother-of-pearl that shines with sudden and mobile light. The carved wood sculptures of saints and sinners, so famous in 17th-century Spanish art, here include works – for once – by a female artist, Andrea de Mena. Her weeping Virgin is as small and delicately carved as it is deeply poignant, shot through with maternal suffering. And hanging alongside the Duchess of Alba is one of Goya’s black ink and wash drawings from Album B, known as the “Madrid” Album, showing a woman standing by the marital bed in the middle of the night. Her husband lies stolidly slumbering as she investigates her white chemise for fleas (or something worse). It is gentle, tender, full of empathy for the put-upon wife. “To Spain I do not go as a plunderer,” wrote Huntington. “I buy no pictures [there], having that foolish sentimental feeling against disturbing such birds of paradise upon their perches.” Instead he bought from auctions and from other collectors; and some of his paintings were commissioned directly from contemporary artists. There is a whole gallery here of sunlit-flickering gardens, seascapes and picnics by the fin-de-siècle Valencian painter Joaquín Sorolla. But what strikes is the searing art of 17th-century Spain: El Greco’s stark Pietà, where the body of Christ and the weeping women are all contained within the embracing form of the Virgin Mary, and his monumental Saint Jerome, naked against a flaring sky, looking with such compassion upon the figure of Christ on the cross. Zurbarán’s extraordinary portrait of Saint Rufina as a pale-faced Spanish beauty, eyes to the skies, dressed in green taffeta against rose-pink silk. Velázquez’s early portrait of the Spanish prime minister Count-Duke of Olivares soars up the wall. An overbearing bully in black silk and regalia, he also has the plump chins and kiss curls of a baby. Velázquez had only recently arrived at the court of Philip IV in Madrid but he already has the measure of this alarmingly powerful figure. But the jewel of the whole show is the most modest of all the works here: Velázquez’s portrait of a Spanish girl, innocent, dark-eyed, aged perhaps around six or seven and so tenderly observed by the artist as he works that one might surmise a family relationship between them, perhaps of grandfather and grandchild. Her clothes are just a rapid salad of marks. But her soft hair is touchingly unruly, his brush stroking a tendril that has come out of place. And he takes his time to paint her grave little face, flawless in the pearly light. This portrait, too, remained with the painter until his death. It is a painting of pure love. Spain and the Hispanic World is at the Royal Academy, London, until 10 April