DEVONtechnologies cannot be expected to test and approve all the various copy/clone/backup applications that people use, in the variety of conditions in which they use them.
That said, I’ll tell you how I manage synchronization issues. 
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I try to avoid like the plague having two or more copies of a database diverge in content by modifying two or more of them differently.
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There are fundamentally very, very tricky logical problems in doing two-way synchronization of databases that have diverged in content. The best solutions to those logical problems will result in making you, the human, ultimately responsible for making decisions, such as in a case when you have edited a file differently within both databases. I hate a case like that. I don’t want to have the software simply choose one of the versions, but then I’ll end up scratching my head trying to figure out what I had been thinking and picking and choosing between elements of two versions of that file. But, as the FAQ warned, attempting to use two-way synchronization software may very well just damage your database.
Personally, I use the Gordian Knot solution. I need to be able to work on my databases whether I’m at home or on travel. My solution is to use a laptop as my primary computer. Problem solved.
OK. Sometimes I may need to work on a different computer. The easiest way I’ve found to do that is to copy my databases to a portable FireWire drive and run them on that drive. My databases are self-contained, so that’s easy to do. Performance isn’t as fast as when I’m running the databases on the 256 GB SSD in my laptop, but it’s acceptable. When I’m finished working in that mode I simply copy the modified databases back to my laptop. (I recently bought a 500 GB 7200 rpm 2.5 inch drive for $129 and stuck in into a case with FireWire 400 connection, so that’s not a very expensive approach. Smaller HDs have gotten really cheap.) It’s also possible to run a small database from a little thumbnail USB drive, although performance won’t be great and it may not be as reliable. Thumbnail drives are great as ‘sneaker net’ devices, though, to transfer files from one computer to another.
Once in a while, in a fit of temporary insanity, I may add new content to a database from a computer that doesn’t hold the master copy of that database. Perhaps I’ve downloaded some files from the Internet, or Imported some files that are on that computer. Rather than running synchronization software (I probably don’t want to copy that other computer’s files to my laptop, and certainly don’t want to synchronize my laptop to it) I’ll open the History tool on that revised copy, identify the recently added content (Date Added), select it (Date Modified sort) and export it to a new folder in the Finder. If I had edited a file, I’ll remember to also select that for export. Then I’ll Import the contents of that folder to my master copy on my laptop. Problem solved. (I’ll have two versions of that edited file in my master database, but that’s OK – I can choose one or the other, or incorporate changes into one and then delete the other.)
I have redundant backups and copies of my databases.
I use Time Machine to backup my laptop. I keep copies of my databases or Backup Archive archives of them on another computer, and I use a portable drive that holds recent Backup Archive archives for offsite storage.
I don’t use online storage of databases or backups, simply because my relatively slow and unreliable satellite broadband connection (a rain shower will knock me offline, whether it occurs locally or at the uplink site in Texas) isn’t sufficiently dependable.
Other users may have alternative approaches that work well for them.
Although the FAQ post recommends against use of MobileMe (because of Apple’s implementation of WebDAV) to store a database, DEVONtechnologies is working on approaches that may allow use of MobileMe in the future. But storage of a zipped database file such as a Backup Archive file on iDisk will work well now.
Some users report success using DropBox or similar approaches to allow synchronization of a database between their desktop and laptop computers. Note, of course, the caution that a database must be closed after use on one computer before it is accessed by a second computer.
There are many backup applications, and most of them are capable of incremental backups of drives (including DEVONthink databases), including ChronoSynch. Eric uses and recommends rsynch. And there are a number that will successfully clone a bootable backup drive (which is a nice thing to have in case of hard drive failure.) I depend primarily on archives created by the Backup Archive script for my database backups, because offsite storage is the greatest security. I like the fact that the filename of an archive contains the backup date. Although that script was removed from pb6, Christian posted a copy of it at viewtopic.php?f=3&t=8948#p41541
Just remember that synchronization is a term that get tossed around with various meanings, and the practical effect of approaches to keeping a database up-to-date can be accomplished in various ways. I try to keep it simple, reliable and as easy as possible.
