After some time with Google, we determine it’s a Polka-Dot Wasp Moth. The ‘wasp’ part is just because of the body shape. It’s a harmless moth.
It’s also a pretty cool looking moth. So, we took a nice close-up photo of it to show everyone what it looks like. The picture below is larger than actual size. The actual moth is maybe an inch in length. For reference, the gray stripe is a line of mortar in our brick wall.
Turns out, this is what happens when the caterpillars that ate our oleander, grow up. The caterpillars are orange with black tufts of hair, which is interesting too (check the Google images for "oleander caterpillar"), but the grown-up moth is unusual and interesting looking.
Those colors have not been modified. It’s an iridescent green bug with white spots and a red-orange tail-end. Interestingly, when I searched online I found that the spot pattern seems to be fairly consistent on these guys.