Chopin Scherzo Connection

I was teaching the other day, and my student MoonKi played the Chopin 3rd Scherzo for me. I love the piece, and was listening to her play it when something struck me. A possible thematic connection between the 3rd a 4th scherzo?

I've played all of Chopin's Scherzi many times, and I've also performed all 4 in the second half of some of my recitals. I had never noticed this connection!

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.