Undoubtedly, Joseph Haydn's Divertimento is a staple of the cello repertoire. Besides the 2 cello concertos, Haydn didn't leave us with any sonatas, so a fancy arrangement for cello and piano is always welcome. The story of Piatigorsky's arrangement is not as simple as one might think.

In the 1920s, the French cellist Pierre Ruyssen was trailblazing pedagogical cello arrangements much like his colleague Louis Feuillard. One of the arrangements for cello and piano was a set of two divertimentos (Deux Divertissements pour baryton) by Haydn. The second of these divertimentos is in D major and in four movements. The movements are as follows:

Adagio (first movement from Baryton Trio Hob. XI:113)
Allegro di molto (second movement from Baryton Trio Hob. XI:113)
Menuet. Allegretto (third movement from Baryton Trio Hob. XI:95, "menuetto" in the MS)
Finale. Vivace (third movement from Baryton Trio Hob.XI:81)

Ruyssen's transcription sticks very closely to the distribution of the voices: the cello takes the baryton part, the piano right hand takes the viola part, and the piano left-hand takes the cello part. Ruyssen is not fanciful with ornamentation and does not introduce too many changes.

In comes Piatigorsky, probably studying the Ruyssen arrangement as a student, or hearing another student perform the arrangement. He obviously saw the potential of Ruyssen's work to become a virtuoso cello piece. Piatigorsky took Ruyssen's arrangement as the skeleton for his own 1944 "transcription," rearranging the order of the movements, dropping the finale, and heavily ornamenting both cello and piano parts. The new order of the movements is as follows:

Adagio (first movement from Baryton Trio Hob. XI:113)  
Menuet (third movement from Baryton Trio Hob. XI:95, "menuetto" in the MS) 
Allegro di molto (second movement from Baryton Trio Hob. XI:113)

Reordering movements was common in the 20th century; we see this in the reordering of Boccherini's famous A-major cello sonata, where cellists commonly play the Adagio movement first (Boccherini had the Allegro movement first) and drop the finale.

It is highly unlikely that Piatigorsky ever saw Haydn's original trios. He was obviously working from the Ruyssen arrangement. His choices of movements and spelling of "Menuet," are 2 of many markers to demonstrate this. The new melody in the Trio section of the Menuet closely resembles the rhythm of the Trio in Haydn's Symphony No. 85. 

Piatigorsky reharmonized the Divertimento in places, so it sounds more like a Kreisler "in the style of" piece than Haydn's original. For example, Piatigorsky goes to a 3rd inversion dominant harmony in m. 2 of the Adagio. Haydn would have resolved it to a 1st inversion tonic harmony in m. 3, but Piatigorsky's ornamentation of the cello melody does not allow for that resolution. But enough about music theory for today.