Question

window.addEventListener('unload', function(e)
{
    MyClass.shutdown();
    window.removeEventListener('unload', /* how to refer to this function? */);
}, false);

Answer 1

Name your function.

function f(e) {
   MyClass.shutdown();
   window.removeEventListener('unload', f);
}
window.addEventListener('unload', f, false);

Edit I think this will work too. Good point Kobi!

window.addEventListener('unload', function f(e)
{
    MyClass.shutdown();
    window.removeEventListener('unload', f);
}, false);