Since we don't know why the hand-crossing was suggested, here is some discussion of how the piece could be played with it, and why.
For reference, the score can be found on IMSLP.
I find no passage facilitated by hand-crossing, so unlikely this was the reason.
The piece is laid out very well, and notated clearly, to be played without hand-crossing. There is some hand overlap such as the transfer from the very first note (RH) to the second (LH) or the alternating chord/broken interval sections where the RH thumb encroaches on the LH chords. But crossing hands in these places makes them no easier to play.
There are three primary passages that allow for some cross-hand showing off.
The most effective is the alternating chords/broken intervals section from mm. 26-29. The chords can be played very staccato with the left hand, which then leaps to play the highest note, before returning for the next chord. The right hand attends to the "inner" repeated notes. This could also be done the other way around, with the right hand taking on chord-leap duties. If you really want to get fancy, alternate roles periodically (e.g., each measure/half-measure/beat).
Measures 14 and 16 can be played as above, but the two measures not being contiguous, and being shorter overall, it doesn't give the audience as much time to realize what's happening and be impressed.
The third possibility is m. 21, in which the right hand can take the single notes and the left hand the arpeggiated groups. However, since the hands are fairly close together and remain crossed the entire passage, the effect is muted.
Free Bonus Answer: For left hand alone
If you really want flair, the way to go is to play it one-handed. Even Wikipedia agrees:1
The work is often performed by left-hand alone.
Much of the piece is straightforward and the fingering intuitive, but a few places require some liberties. For a detailed explanation, see (How) Can CPE Bach's Solfeggietto (Solfeggio) in C Minor be played left-hand alone?
1The Wikipedia article references a source no longer available. Here is the Wayback Machine version: http://web.archive.org/web/20180726135755/http://www.piano-lessons.net/news_item.php?id=59