In Italy they're called girasole/girasoli (singularplural), which means the same thing. That doesn't oblige them to do so.

When I said I'd not observed the immature buds turning, I wasn't denying that they do so. I would have to actually invade a field and get up close to see immature buds, and the farmers here don't like that kind of thing. When I see the green flowerheads, already fairly big before the petals appear, they're already locked in the eastern-facing direction.

I liked Cortona. It is just as much visited as Montepulciano, which is extremely touristy.
I much preferred Cortona.
Pienza was lovely too.

I've visited zillions of hill towns and found them all to be neat and just about as nice or whatever as others - small hill towns anyways. Larger ones like Perugia, Siena and Assisi have more unique attractions perhaps and lack the natural ambiance of a smaller hill town.