So I did a little reading. For rhizomatic plants, the horizontal vine is actually the stem. It's basically one big plant growing horizontally underground. That's why if you don't pull it all, you're just plucking shoots off the stem. And like most plants, if you cut up the stem and plant it, each section will grow a new plant.
Of course, if you completely shred it into very fine pieces, it won't be able to regrow. But it can come back from fairly small ones. You might think you've gotten it all, but it's just regrowing until it erupts again.