Well, that hinted me in the right direction!
First, I had iCloud Drive turned off in my iCloud sync settings on my phone. So I turned that on. (I turned this off before I had paid iCloud to reduce storage usage in my free iCloud account.)
Then, I toggled Orion in my iCloud sync settings on both devices off and on again.
Finally, I toggled Orion sync within the Orion's app settings off and on again.
Now my bookmarks are syncing as expected 🙂