Its true an SPL competitor isn't going to invert his subs for fear of losing that slight bit of air displacement caused by the frame/motor blocking some of the radiating cone surface area (very little). But in a daily system its fair to say either mounting style will give the same results in regard to output. Inverting a sub does tend to offer slightly better cooling capabilities, but also adds to the possibility of audible motor noise from the driver. All are considerations that should be thought of when designing the system, but really it comes down to mostly just for looks.
I believe what was meant about saying you had to reverse polarity when inverting was that assuming the polarity was set correctly when the sub was mounted traditionally, when inverting the sub you must reverse polarity. But yes like squeak said, inverting a sub does not necessarily mean its polarity must be reversed in relation to the other speakers in the system. The sub may sound better out of phase with the front stage, for example. But this comes down to the difference between relative phase and absolute phase. If you research and learn what those two terms mean, you'll then understand the whole phasing issue and what happens when a speaker is mounted inverted.