How can I keep copies of my iCloud documents in Dropbox? More and more of my favorite apps are beginning to include support for document syncing via iCloud.

I like the idea of using iCloud to store my documents, but for several reasons I'm not ready to switch over completely from Dropbox.

Is there some kind of an iCloud-Dropbox bridge or other way of syncing documents between iCloud and either Dropbox directly or the local filesystem of my Mac? Ideally it would work both ways, pulling from iCloud and pushing into it.
I am not absolutely certain, but I believe this is impossible.

For one thing, there isn't iCloud support for simply syncing files. Apps are expected to keep their synced files in their own folders (and are restricted to accessing this folder alone). This means that there's no way to have an iCloud document accessible my multiple apps.

For another thing, even if you were willing to concede and only sync, say, Pages documents with iCloud and Dropbox (hypothetically), Pages for iOS documents are very different from Pages for Mac documents. Here are two screenshots, one of the inside of a Mac Pages document and one of an iOS Pages document, respectively:

 

Post script: hypothetically, though, if you could figure something out, it would probably involve somehow linking ~/Library/Mobile Documents with Dropbox.