I know it's not cool to answer your own question, but another possible solution occurred to me last night, and it only requires 1 new line of code to be written (sort of).
The first part of the solution has already been implicitly suggested by many of you. Modify the JavaScript to write to a cookie:
function toggleMenu() {
var navigation_pane = document.getElementById('site-menu').style;
if ( navigation_pane.display == 'none' )
navigation_pane.display = 'block';
else
navigation_pane.display = 'none';
document.cookie = "menu_vis=" + navigation_pane.display; // +1 line of code
}
Now, what are the possibilities if your CSS file just so happens to be a PHP file in disguise? my_css.php
would look something like this:
<?php
header("Content-type: text/css");
?>
#site-menu {
display: <?php echo isset($_COOKIE['menu_vis']) ? $_COOKIE['menu_vis'] : 'block'; ?>; /* line of code modified, but not added! */
}
Tested this morning, and it works.
I find it a neat solution, because it means that I don't have to bend my PHP or HTML design around any presentational concerns.
--
I appreciate that there are more "encompassing" solutions out there. If I was a better JavaScript developer, (or made use of jQuery or the like), I could build more complicated classes which could then be applied more generally to other HTML elements. I may come back to investigate such solutions later, but that's just not where my project is at the moment.
Thank you everyone for all your replies. I wouldn't have found this solution without bouncing these ideas off you guys.