Mystery solved!

I've finally discovered the name of my mystery plant. I learned another common name for it - "tubeflower" - searched for it under that name.

It's Clerodendron indicum. Somebody in comments guessed "Clerodendron" and they were right. Another common name is "turk's turban" for the reflexed-backward petals that look (well, it's a stretch) a little like a hat; also "skyrocket" and "bowing lady."

Native to China and Southeast Asia, it grows in Southern states of the US; a perennial, it can be invasive. It's an old-fashioned flower; my neighbor says it was growing on his property before he came there.

Its seed pods are maturing right now, I hope to gather some seeds.