These have been one of my favorite flowers for a long time. I grow them too. From what I understand, they grow best in ditches to catch rainwater run-off, and the sloped land also helps the long stems stand upright. The little bulbils probably roll downhill and into streams, where they can grow on the edges. Sometimes they are called Turkish Caps. Usually I provide something for them to lean on because the stems will snap easily if gravity pulls them down, and they sometimes do not grow back if they snap early.
Yes, we also sometimes call them Turkish caps.
It is strange to me why they are called that.
Thanks for the additional information.
This lily is really beautiful.