Cadenzas are unaccompanied flights in a concerto that play on its themes and show off a virtuoso’s mettle. In the 18th century performers often improvised. But increasingly composers wrote down their cadenzas. Sometimes they were written by other composers.

Ms. Grimaud and Mr. Abbado got together in Bologna, Italy, on May 29 for several days of rehearsing, performing and recording with Mr. Abbado’s Orchestra Mozart. The aim was to produce a Deutsche Grammophon disc of a vocal work and two Mozart concertos, No. 19 in F and No. 23 in A. Mozart wrote cadenzas for both. But Ms. Grimaud arrived with another cadenza for No. 23. It was by Ferruccio Busoni, an Italian piano virtuoso and composer active in German-speaking lands in the late 19th century and early 20th. Vladimir Horowitz, in his recording of the piece with Carlo Maria Giulini also for Deutsche Grammophon, played the Busoni cadenza. Horowitz fell in love with it after declaring Mozart’s cadenza “foolish” and “too thin,” according to David Dubal’s book “Evenings With Horowitz: A Personal Portrait.”

Ms. Grimaud did not go that far. But she said that Mozart’s cadenza for the work was “not the most inspired.” She said she too had fallen in love with the Busoni version when she was 12 or 13, after hearing that Horowitz recording, and always knew she’d play it. “It’s brilliant,” she said. “It’s inspired. It’s very imaginative in the way it treats the material.”

The Busoni also has a lush, Romantic sound with hints of Brahms and Liszt — composers with whom she has made her mark.

Mr. Abbado said he preferred the Mozart cadenza, but they rehearsed, performed and recorded the concerto with Busoni’s. At a touch-up recording session, Mr. Abbado asked Ms. Grimaud to play through the Mozart cadenza.

“I said I would not seriously consider doing that for the recording,” Ms. Grimaud recounted. To begin with, she had not practiced the Mozart cadenza. But Mr. Abbado insisted, and Ms. Grimaud retreated to her dressing room to work on it. She came out and ran through it twice, and the recording engineers kept the microphones on.