Update:
After a recent update (sudo apt-get update && sudo apt-get upgrade), the partition which was being mysteriously mounted on /media/DATA is now being mounted instead on /media/$USER/DATA, unleashing all types of hell in the process.
Adding an entry in /etc/fstab to mount the partition back on /media/DATA appears to sidestep this issue. Nevertheless, it would be nice if it was possible to know how Ubuntu handles these partitions instead of relying on faith and blind luck.
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