Can you install two versions of an app on the same iOS device? Is it possible to install two versions of an app on the same iPhone and, if so, how?

The scenario would be that you have purchased an app, and a recent update has changed some of the features--not necessarily for the better or worse, but enough of a change that you'd like to still be able and use the older version at times. You could downgrade from a backup, but it'd be nice to have the new version at times as well. Is it possible to have two versions of an app be 'seen' as different apps so that they can both be transferred to the iPhone at the same time?
It's not so much the name -- at least not the visible one, as an upgrade can change the visible name -- but it's the bundle identifier inside the app that identifies it and allows the OS to know it's supposed to perform an upgrade. The developer has to keep the bundle identifier the same in order to actually provide an upgrade and not a new app, so no, you're out of luck.*

*Note that there may well be some way to do such a thing if you jailbreak your device, but I have no firsthand knowledge of such.