OK I narrowed down the problem, and found a fix...
What I did was uninstall IE9, which reverted back to IE8. I then tried to open the shortcuts on my desktop, and noticed that there wasn't any application ascociated with the shortcut. Turns out the shortcut has a "website" extension as opposed to "html" or similar. IE8 doesn't understand "website" extensions.
So I re-installed IE9 and once again the links opened IE9 64-bit. I then went into the control panel to change file associations, found the "website" extension and simply re-selected Internet Explorer as the app to use. This time the links open IE9 32-bit.
Is this a bug in the IE9 installer, or just some weird glitch on my system? Not sure.