The native resolution of my LCD is 1680x1050. When I go into hardware options and pick "1050x1680" everything on screen is rotated 90 degrees. Is this just some weird mistake?

I'm not sure but It looks that everything is allright. :) You have selected 1050x1680 not 1680x1050. You can try to ratate your screen in the drivers setup.

I would gladly click on 1680x1050 if it showed up in the selection box.

Heh, strange. :) I have no idea then, sorry.

Strange. My laptop's 1200x800 and I didn't have any surprises like that. ATI X600 chipset.
Any chance you've got a driver that needs updating? (Totally guessing here)
Does it seem to work right at lower resolutions? 1024x768? (I know it's not native for your LCD, but there weren't 1680x1050 LCDs in 2001)

The game clearly has an option for the 1050x1680. All I want is 1680x1050. Is that so odd?
And yes, I've tried lower resolutions. They look awful on my LCD.
Driver's updated too. Since you don't have any problems with 1280x800, are you selecting 1280x800 or was the only option that showed up in the graphics settings 800x1280?

I have two 1200x800 selections, 32 bit and 16 bit. None have x/y inverted. The reason why I asked about the driver, and you haven't told us what card you have, is that the software is obviously asking the card/driver what resolutions it can support and appears to be screwing up.
If you have an ATI based card, you might try one of the third party drivers like omega (For more info, look here - http://www.firingsquad.com/matrix/blog.asp/62001/311 ) I don't actually know of a third party option for nvidia chipsets.
Just doin' what I can to help!

It's an Intel GMA 950 with the latest drivers and I run 1680x1050 in Windows.
I asked tech support about it but they told me the solution was to use the monitor rotating options in my drivers, but using them just makes the game close.

You could try disabling any auto-adjust type options for your drivers/monitor.
It could be that your display is being *too* clever and auto-rotating to what it thinks would fit the resolution.

In all reality, just run it at a lower res and don't worry too much about it: it sounds like your monitor/card are looking at the resolution, and the game, and going "OK, how the Hells do I make this fit? Uhm... that looks right. I guess. Feh. Weird-ass resolution, anyway." and giving you the invert.
As for a FIX, everyone's already said what you should do, which is manually adjust it from your monitor or dashboard vid card settings, but if that just crashes out the game then, again, just run it at a lower res.