With jquery you could do to sth. like this, which will simply switch classes.
$('.showhide').click(function() {
$(this).removeClass('myclass');
$(this).addClass('showhidenew');
});
If you want to switch classes back and forth on each click, you can use toggleClass, like so:
$('.showhide').click(function() {
$(this).toggleClass('myclass');
$(this).toggleClass('showhidenew');
});