These were the steps:

1. Created a content document called style.xml, with three links, one for each style (the page is here). Actually, they were originally links, but for various reasons (mainly oddities in the behaviour of Safari and Konqueror with anchor tags which have no href attribute), I ended up making them <span> tags with an onclick event.
2. Created a pipeline just for that page, for testing purposes, and implemented the branching structure for that pipeline.
3. Created a couple of JavaScript functions in the utilities.js file. One just sets a cookie (adapted from this page). The other does this:

   function SwitchStyle(NewStyle){
       SetCookie('PageStyle', NewStyle, 365, '/', '', '');
       window.location.reload();
   }
   

4. Set the onclick event in the <span> tags on the page to call the SwitchStyle function.
5. Tested the page on Firefox, IE6, IE7, Opera, Safari, WebKit and Konqueror, and tweaked until it worked the same on all the files.
6. Worked through all of the relevant matchers in the stylesheet, adding branching blocks as appropriate, and testing as I went.

Everything seems to be working well. IE's lousy CSS support makes all three themes less attractive on that browser, but that's life.

Now I need to get the text on the style-switching page translated, before I can add it to the site menu.