Example 1 (click to enlarge)
Example 2 (click to enlarge)
The two works date from 1839 and 1843 respectively. Number 3 is in C# minor, while the fourth is in the relative E Major, so they also share the number of sharps.
The descending double-octaves of the third scherzo (example 1) are a mirror image of the ascending chords of the fourth (example 2). As soon as the 3rd scherzo modulates to E Major, look at the five indicated notes (Example one, first line, bars 10, 11, 12, 13, and the first bar of line 2) : it’s the opening theme of the Fourth Scherzo!
I don’t know if it was intentional, but it is hard for me to believe it wasn’t. I think Chopin started out the 4th scherzo as a transfigured version of the 3rd and moved on from there. Two works sounding as different as can be, and yet very closely connected.