It turns out that Windows XP Service Pack 3 broke the ATI software. Because I wasn't using this feature at the time, I missed out on all of this drama -- as well as the opportunity to download the updated device drivers. Now the device drivers (if they ever did exist) are buried in some archive that I can't get to or find after hours of Googling. I could find many drivers for the ATI cards, and even the ATI Radeon Mobility cards, and even the ATI Radeon Mobility X(number) cards -- but not the X1400.
I think this is because Dell and Lenovo (and probably other OEMs) packaged this card with their PCs and didn't want people downloading device drivers from ATI that would cause problems because they would have to field the tech support calls. Which is what this ATI site confirms.
Anyway, I searched and searched and couldn't find any drivers, until I found this:
http://www.hardwareheaven.com/modtool.php
That's right, someone actually wrote a tool to modify the ATI drivers so they would work with other cards. Now I can download the drivers meant for the 22 other models and they will work with my card, which is conveniently left out of the list.
And it works! I now have a laptop working with an upside-down screen!