For Mr. Cogitore and Ms. Dembélé, that something has been clearing out the piece, rendering the stage spare and lucid, and permeating it with street and club dance — not just krump but also flex, vogue, break, electro and more — to empower the work’s “others” to represent themselves on their own terms.

Performed in front of a largely white audience on Sunday afternoon, this was still a brand of exoticism. But Mr. Cogitore and Ms. Dembélé have shifted the work’s power dynamics substantially toward egalitarianism.

They’ve done that without stinting on the wonder-inducing spectacle that was the Baroque’s reason for being. A carousel turns; vitrines slowly levitate; cheerleaders wield sparkly pom-poms. An explosive bout of break dancing brings to life the eruption of a Peruvian volcano.

The dancing and music share a ferociously polished extravagance. As the soprano Sabine Devieilhe, who plays several roles, delivers a solemn prayer to the god of marriage, Calvin Hunt, a flex dancer and a member of Ms. Dembélé’s Compagnie Rualité, gives a stage-filling solo as sublimely floating as her singing. Leonardo García Alarcón leads his Cappella Mediterranea ensemble in a vibrant performance, propulsive and tinged with the fermenting tang of period instruments.

Baroque pieces are usually performed in the Paris Opera’s ornate, relatively intimate Palais Garnier, but this “Indes Galantes” both musically and theatrically fills the 2,700-seat Bastille opera house. And while “La Traviata,” like other standard-repertory works, would generally be performed at the Bastille, Mr. Stone’s staging felt faceless even in the cozy confines of the Garnier.