Reverse shipping addresses

I didn't know about that. I've always seen the checkbox based on the billing address on ecommerce solutions. And that's normal since you don't always have a shipping address (for virtual goods for example) but you always have a billing address.

Anyway, I think that the simplest will be to change text in the translation files to replace shipping by billing and vice-versa and then, you'll also have to edit your shipping plugins code to use the billing address instead of the shipping address.

For example, in UPS you just change the line:
$cart->loadAddress($null,$order->shipping_address->address_id,'object', 'shipping');
to:
$cart->loadAddress($null,$order->billing_address->address_id,'object', 'shipping');

Other solutions like changing the checkbox system will be much more complex than this.