You could create a form (with the dropwown in it) that is automatically posted every time the state of the dropdown is changed. Use Javascript to catch the event. In addition, use Javascript in order to post different values based on the different sorting selections. If you submit the data using the POST method then you'll have to extract it using the ezhttp template operator. If you're using the GET method then you'll have to incooperate the desired sorting method into the URL. This is possible by making use of the "view parameters" mechanism: http://www.example.com/your_page/(sort_method)/alphabetical. Extract the variable like this: "$view_parameters.sort_method". Based on the extracted variable you can then do different sortings in your fetch.
Isn't that functionality of 3.4 only? I would love to see it appear in 3.3-4 or so, this reduces the manual patching after an upgrade of 3.3-stable for me :-)