Continuing with absolute difference of a 2-digit number with its reverse (Kaprekar's Algorithm for 2-digit numbers)

Question

Let $ab$ be a two-digit number($a \ne b$) with its reverse as $ba$, it is simple to check that $|ab-ba|=9|a-b|$.

Interestingly, if we continue as $$ |ab-ba|=cd \to |cd-dc|=ef \to |ef-fe|=gh \to |gh-hg|=ij \to |ij-ji|=kl \to …. $$ we will encounter 9: soonest, sooner or soon, wherein $cd$ or $ef$ or $gh$ or $ij$ or $kl$…will be 9. And then we get a periodic string of $(81,63,27,45):$ $$ 9, (81,63,27,45), 9,(81,63,27,45), 9, ...$$ This result is inspired by Kaprekar's (algorithm) constants 6174 and 495 for 4-digit and 3-digit numbers, respectively.

So is 9 for 2-digit numbers but in a different way.

Any reference to this observation or any comment is most welcome.

One may see the links for the Kaprekar constants 6174 and 495:

https://en.wikipedia.org/wiki/6174_(number)#:~:text=6174%20is%20known%20as%20Kaprekar's,at%20least%20two%20different%20digits.

https://en.wikipedia.org/wiki/6174_(number)#:~:text=6174%20is%20known%20as%20Kaprekar's,at%20least%20two%20different%20digits.