Inverting your sub will increase the box volume giving you the full chamber. This usually doesn't do much because the subs displacement (depending on what the enclosure size is) doesn't affect it that much. For example, an 18" sub usually takes up around .25 cubic feet of displacement. In a 7 cube box to 7.25 cubes you won't notice much, if anything.
The problem with running inverted is mechanical noise (especially on low notes, mostly below 35hz). When the sub is mounted "normally" in the enclosure a lot of that mechanical noise is muffled. If you wanted to show the sub's ass, I would design the enclosure to have some form of plexi or something so you can see the sub. Much more practical than inverting it, IMO