That's correct, Ken. The UPS acts as a surge protector so it provides protection as well as battery backup. I contacted Belkin and they advised against plugging one unit into the other.
I have only the equipment I feel needs battery backup for proper shutdown plugged into the UPS. The power center has a few features the UPS doesn't (for example, switched and delayed outlets), so I find them both useful.
"People generally quarrel because they cannot argue." - G. K. Chesterton