Is there any way to save the order of spaces in Mission Control?

What Jay said in a comment... Uncheck the second box here:

It's possible to assign apps to spaces in OS X 10.8 Mountain Lion, but it does not work for full screen apps (bold italic emphasis mine in the Apple support document below).

Decide which spaces to use with particular apps

If you work in multiple spaces, you can assign apps to particular spaces.

1. Press and hold an app’s icon in the Dock. You may have to first open the app from Launchpad to see its icon in the Dock.

2. From the shortcut menu that appears over the icon, choose one of these:
   To have the app open only in the current space, choose Options > This Desktop. From now on the app will open in the current space; if you are working in a different space, the app’s space scrolls into view.

   This setting doesn’t apply to full-screen windows, which always appear in their own separate spaces.

You may have a better solution in the paid app TotalSpaces.