You can arbitrarily reorder your photostream (not just sets, your actual photostream), and you can check the organizer to filter your photostream for only public photos.
The key is "upload date" which is editable, and determines the order.
If you don't have a tool for it, go to your set in the organizer, sort the set as you want, go to the first photo, open the edit modal dialog, and start stepping through the photos setting the upload time to 1 second after the last. When you're done, the photo stream itself will reflect the custom order.
Sets already reflect your custom order with no extra work. Some Flickr tools (of which there are hundreds) support reordering the photo stream, using this technique under the hood.
Since everything on Flickr has an API, you can do most anything you can imagine.
wow, that's awesome. I hadn't realized they had made upload date writable! This has some interesting possibilities. It is somewhat too bad that changing it will wipe out the original photo upload date, though - for photos which don't have the date taken in the exif, there will be no time history of the photo left.
You can arbitrarily reorder your photostream (not just sets, your actual photostream), and you can check the organizer to filter your photostream for only public photos.