I find this clear now, like so (argument discovered by others such as @NoahJSnyder and @MtgJulian; I am only reframing or rewording it):

https://twitter.com/littmath/status/1769408478139785497

Consider a random walk in which one takes equally likely steps of one unit up or one unit down, but with different distributions of speeds. (E.g., maybe up steps take one hour, while down steps have probability 1/2 of taking 2 hours, 1/4 of taking 3 hours, 1/8 of 4 hours, etc).

@llllvvuu @littmath @NoahJSnyder @MtgJulian Summarizing the proof you all devised:

A "tied" string has equally many HHs and HTs. A "flippy" string is tied of size > 1 starting and ending with H, with no such proper prefix. Reversal is a length-preserving involution exchanging flippy strings starting with HH vs. with HT. @llllvvuu @littmath @NoahJSnyder @MtgJulian Say S has base P if S is of the form PHQ, PH is tied, and HQ has no flippy prefix. As there are equally many Q of a given length starting with H vs. T, our above involution tells us there are equally many strings of a given length whose base is followed by HH vs. HT.

I condemn Hamas killing 1400 Israelis.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans.

I condemn Israel killing 1400 Gazans. I condemn Israel killing 1400 Gazans.