You wouldn't think it would be so difficult to identify a plain white fuzzy caterpillar.
But it is.
I believe it is the white form of Spilosoma virginica - the Virginian Tiger Moth, a.k.a. Yellow Woolybear.
On the BugGuide web site, there are just a few references to the white form. The other forms look so different!
The adult moth is fairly attractive, as moths go.
The Alabama Cooperative Extension System (ACES!) offers Identifying Caterpillars in Field, Forage, and Horticultural Crops.
There is an extensive key that goes along with the publication, but having neglected to count the caterpillar's legs, I was lost.
At first glance I thought this might be one of the many stinging caterpillars present in Alabama, which is part of the reason I wasn't too keen on leg-counting.
But after looking through the pictures, I don't think so. (That page would be so much better with thumbnails.)
I have a strong "better safe than sorry" policy when it comes to things that sting or bite. I guess I should have tried to get the caterpillar to climb onto a stick for further ID purposes, but there was Jasmine to consider. She tends to have a strong "stick your nose right on it" policy.
Edited to add:
More info on stinging caterpillars is here and here. For specifics in your area, try googling the name of your state or province along with the words "stinging caterpillars".