How to allow Readers to Select Post Order?

What you need here is Ajax. You will need to read about how to use WordPress Ajax (link).

When your visitor click on select field to choose sorting, you need a JS function to handle the data about how to sort and where the visitor in, then a js function to make ajax request to server.

At server here (I mean your theme function) you will need a function to handle request and send the response to client. The handler function here is a WP_Query loop to loop your post according to the order that your visitor chose.

Sorry, I don’t have enough time to provide you the code, but I think for a scripter/coder/developer, the direction is enough.

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