After we got a few brief glimpses through our binoculars (a birder's best friend -- well, those and a good field guide), it flew away and we flew to the computer.
Our friends at Bird Web provided the solution. On the pileated woodpecker page, there are several photo suggestions for birds that look like it. Tiny Girl pointed to one. "That's it!"
Typically, Cornell's All About Birds site is my go-to online resource, but the only two similar species they suggested were the red-headed woodpecker and the American crow. (What?? I think I can tell the difference between a crow -- which we see all day, every day -- and a pileated woodpecker!)
But it's good to have a few tricks up one's sleeve, so I'm glad I found Bird Web.
Photo courtesy Wikimedia Commons