It seems that there is no way to move/copy test cases between projects properly.
The API for get_case doesn’t even respond with precondition or step information, let alone attachments.
I tried exporting the test cases in XML and re-importing to the new project. While images embedded in the steps continued to work (as they were pointing to a global resource), the XML did not contain information about attachments.
Is this latter problem going to be looked at at some stage?