This is one of my favorites. I did at least one very “out of the box thing” by changing the rhythm of a few measures from 6/8 to 3/4, which is a rather modern thing to do, but Scarlatti was a very “out of the box” composer. This is also usually played faster. In my view modern pianists put too much emphasis on speed and not enough on shape, accents, contrasts and creativity.
“K” stands for Kirkpatrick. “L” stands for “Longo”.
Kirkpatrick produced an edition of the sonatas in 1953, and the numbering from this edition is now nearly always used. Previously, the numbering commonly used was from the 1906 edition compiled by the Neapolitan pianist Alessandro Longo