I'm not sure where the image is from. It's just a place-holder. I think I figured out how to fix the problem, but I still don't know why I couldn't get it to work the way I had it.
I did have a rounded-on-both-sides green image on the "a:link," aligned to the left of the right button and the right of the left button. And I had a left arrow image on the ".leftbutton span," aligning to the left side. And I had a right arrow image on the ".rightbutton span," aligning to the right side.
I changed the design to use 4 images instead:




…and I used an unordered list instead of spans.
It seems that WinIE doesn't like sliding door where the top image is right-aligned.