These words have a remarkable form of self-symmetry:
ANDROCENTRIC CONDESCENDED EXTRATEXTUAL RESETTLEMENT SUBMERSIBLES
What is it?
It is astounding that @GarethMcCaughan solved this puzzle with so little information. I was fully prepared to drop successive hints for days until someone eventually solved it!
I was not only curious to see if anyone could solve it, but I was especially interested to see how they would explain the answer to a general audience. Again, I think Gareth's explanation is as clear and efficient as can be.
Here is more discussion (as well as a cool visual) for those who are not mathematically inclined:
I was exploring what happens when you take every other letter in a word, every third letter, every fourth letter, every fifth letter, etc. Of course, you quickly reach the end of the word, so in order to keep it fun and use all the letters, you have to wrap around to the beginning again. Like Pac-Man.
I wondered, when you take every third letter, or fourth letter, or fifth letter, do you ever get a new word out of it? It turns out, sometimes you do. For example:
— Starting from the T in the word THREADS and taking every second letter (remembering to wrap around when you reach the end) gives you TRASHED
— Starting from the T in the word TRASHED and taking every fourth letter gives you THREADS
— Starting from the P in the word SPRITES and taking every fourth letter you PERSIST
— Starting from the second S in the word PERSIST and taking every second letter gives you SPRITES
And that's about it. There aren't many other interesting examples, other than very short words. I figured it wouldn't make a very challenging puzzle.
But then I saw something unexpected and fascinating: There are over 100 words that yield themselves again! And the longest of these words are 12 letters long, giving solvers plenty to work with in an enigmatic puzzle.
It is particularly delightful because you get the same word back again, but the letters are now in a different order! Only, you can't tell that they've been rearranged.
If you're a visual person, you might appreciate this graphical representation:
As Gareth pointed out, for each of the 12-letter words I listed, it happens that you need to take every seventh letter. But there's no reason why this needed to be so. It could just as easily have been every fifth letter, for example.
(But it wouldn't work to take every third or fourth or sixth letter. To say more gets us deep into modular arithmetic, congruence classes, relatively prime pairs...)