AN inverted woofer, a woofer mounted basket-out of the box, is said to have better sound quality so I've heard, less resonance off of the basket. It also allows for a smaller enclosure because you don't have to account for motor displacement.
An un inverted woofer is mounted in the enclosure cone facing out. THe basket is inside of the box. YOu have to account for motor displacement, and also you have to run wires from the inside of the box out in order to power your woofer.