As well as adorning the ranks of the Royal Ballet’s principal dancers as one of its most compelling and versatile artists, the Danish-born Johan Kobborg has done the company another big service — drawing on his native classical tradition — by staging productions of ballets by the 19th-century great Dane August Bournonville.

His exemplary rendition of La Sylphide (1836), the earliest extant classic of the Romantic era, new last season, has just returned to the repertory; this time, it is paired at some performances with the exuberant Napoli Divertissements (1842).

Kobborg’s production augments the Divertissements’ familiar pas de six, solos and rousing tarantella climax with an additional ensemble from an earlier part of the ballet. The result is an exhilarating display of Bournonville’s sunniest dances — fleet, buoyant, lilting, uplifting. The cast of 22, mostly on good form (just a few hints of underrehearsal), coped imperturbably with some vagaries of stage lighting at Tuesday’s premiere, which threatened an uncalled-for dusk over the intended sunlight scene.

Marianela Nuñez, Laura Morera, Mara Galeazzi, Belinda Hatley and Lauren Cuthbertson all shone in their skimming dances. Leading the men, the young virtuoso Steven McRae again proved his worth, with his speed, precision, ballon and authority, dancing Bournonville’s brilliant steps as if born to them. One improvement might be to make a bit less of the tarantella’s tambourine-playing — it is rather relentless.

The alternative opener to this Covent Garden double bill is Ashton’s dazzling Rhapsody (1980), the choreographer’s last important work, to Rachmaninov’s Paganini variations. This was excellently danced by the cast of 14 on Monday, doing full justice to the bravura and intricacy of the steps — and, in Leanne Benjamin and Carlos Acosta’s partnership, to the glory and enchantment of the ravishing pas de deux. The focal male role around which this ballet was shaped is majestic, godlike, fiendishly virtuosic, and Acosta is its finest exponent today (though he does not efface memories of Baryshnikov, who created it).

Back to Bournonville and La Sylphide: it’s a lovely stage spectacle in its evocation of 19th-century Romanticism, with atmospheric designs beautifully lit (no problem this time), and in Kobborg’s meticulous preservation of authentic style in dance and mime. The company has taken to its liveliness (the Scottish dances), charm, magic and poetry as if brought up on it — but, before Kobborg, this archetypal classic was never in the Royal’s repertory.

As the captivating, capricious and ultimately tragic Sylphide, both Tamara Rojo and Alina Cojocaru dance with suitably weightless delicacy, but Rojo’s interpretation has more depth, her death scene heart-rending. It is a pity that, due to injury, Kobborg has so far not danced the role of James in his production. Federico Bonelli is, however, superb: handsome, his acting intelligent, his dancing immaculate. The evil witch, Madge, may be played by man or woman. Gary Avis (on Monday) was not as convincing as the veteran Danish artist Sorella Englund (Tuesday), Kobborg’s staging collaborator, who is subtle and mesmerising, every gesture dramatically to the point and steeped in a Danish tradition.

It was felicitous to see La Sylphide in near conjunction with the second-earliest Romantic masterpiece, Giselle (1841), as danced by English National Ballet in a revival of Mary Skeaping’s 1971 production at the Coliseum, the first time they have shown it in London in 15 years. Skeaping’s version is, like La Sylphide, carefully in the spirit of the old style.

In both works, we have super- natural beings: an ensemble of ethereal women in white, dancing lusciously in the forest — an icon of Romantic ballet. Here they are chilling ghosts, out for vengeance on any man. While beautiful (the corps de ballet is very fine), they are splendidly menacing. You thrill at them as the frame for the superb artistry of Agnes Oaks and Thomas Edur (who I saw) as the tragic lovers. The old ballet has lost none of its power.