I see what you mean, but got bad news.
There's a third roll that's on the left handle which is the one you take from, the handles were just folded up when I took this photo.
But you gave me a vision; imagine it like a VCR or tape deck, the paper goes from one roll to another and you spin them to wipe, your butt being the write head for the "tape", a trail of shit getting "written" to the tape.

Like an eggbeater? Sounds kind of risky, brb.