I finally remembered that I wanted to make a serious effort to find out what the two vines are that grow so prettily in the labyrinth.
…with its delicate little stems and beautiful, fern-like leaves, is actually a monster: Lygodium japonicum, an invasive climbing fern that—from the photos in that second link—is every bit as bad as kudzu or ivy. So far, it has shown no inclination to take over anything other than the wire structures I have provided for it.
…is Clematis paniculata, also known as Sweet Autumn Clematis. It is also invasive, and I have to say that I have found a strong cable of vine running through the ivy. I took the remaining seeds and planted them over by the fence in the hopes that I can convince the plant to take part in my eventual privacy fence. I will harvest more seeds next fall.