Critical subwoofer tip

While it would seem that getting the phase to match the mains is ideal, theoretically this isn't necessarily going to result in no peak or notch at the crossover point.  Not unless the slope of the cutoffs for both are the same.  For a mains woofer, in a cabinet, this isn't actively controlled and is prob more gradual than a sub.  Perhaps this is why best sound and SPL may be something other than zero.  It mitigates that complex interaction, peak or notch.  One of the advantages to wiring speakers to a (some) sub's speaker output and using its high-pass.   Also, distance from sub to listener vs mains to listener further makes this issue more complex.   Adjustable phase is an aid for both issues.  Good stuff on this (in discussions of tweeter to woofer xo designing) at Linkwitz Lab site.  

Fun little story...  I was helping my brother set up and dial in his 2 Sumiko S9 subwoofers. I played around with them for a few hours and just couldn’t get them to sound right. Turned out that I had one of the subs 180 degrees out of phase and it was pretty much cancelling out all of the bass from the second sub.

How to set phase correctly:
Since it's easier to hear a null than a peak do this. Reverse the polarity of one speaker (swap + and -) and play a test tone from 20hz to 200hz. Next, from the listening position, ask someone to adjust the phase so you hear a DIP in sound level at some point during the test tone. This creates a null point. Swap your speaker cable back and you're done!

What does a phase issue sound like?
Simply put, there will be an audible dip in the crossover range. Again, only IF there is a phasing issue, will this be noticeable. Importantly, a phase induced dip may lead you to set the sub volume level too high causing, "bloated bass."

Back to the OP's point: If one believes inverse speaker polarity causes a drop in low frequencies, then one must also believe phase of a subwoofer vs. main speakers may cause a similar drop. No, polarity and phase are not the same at all, however, for demonstration purposes, the drop of sound level IS similar.  

Subwoofer phase adjustments are simply (very small) timing delays applied to ONE of two speakers attempting to producing the SAME frequencies. A subwoofer phasing problem is a timing issue.  A few posts stated the problem well: AT THE CROSSOVER POINT of the main speakers to the subwoofer, phase matters. The goal is to slightly delay the subwoofer's sound wave to match the main speakers' phase and have these competing signals reach the listener simultaneously. How much it matters depends entirely on your specific set up. (Eg. room dimensions, LF capability of the mains, location of the sub to listener) WHERE the sub is located in relation to the main speakers affects phase. Specifically, subs located close to mains = little phasing problems, whereas, subs positioned CLOSER to the listening position and FURTHER from the  main speakers = larger phasing problems.

As was also pointed out, subs often reach up to 100 or 200hz and phasing issues become more noticeable, complicated and real. I agree with MC that under a certain frequency it cannot matter.  

Lastly, one cannot "invert the phase." This makes no sense. Adjusting the phase 90 degrees, 180 or 270 of a subwoofer is adding a small delay in the timing of the sound wave. There is no way to do the opposite and speed the sound wave up, hence no such thing as, "inverted phase."
Smooth LF augments all music performances and movies, at loud volumes or quiet volumes. LF done well is glorious.

Not unless the slope of the cutoffs for both are the same.

Not true. One can achieve near perfect integration with a main 1st and 3rd order sub with polarity inverted.

Slopes must be adjustable. Ideally on both the main and the sub. They do not have to match.

Subwoofer phase adjustments are simply (very small) timing delays applied to ONE of two speakers attempting to producing the SAME frequencies.

Not unless phase adjustment is done with a delay or by physically moving the speaker.

Lastly, one cannot "invert the phase." This makes no sense. Adjusting the phase 90 degrees, 180 or 270 of a subwoofer is adding a small delay in the timing of the sound wave. There is no way to do the opposite and speed the sound wave up, hence no such thing as, "inverted phase."

A typical phase control does not add delay. It shifts the phase relationship in the speaker range. If one inverts polarity, the phase shifts. The timing remains the same.

Adding a sub perfectly to an existing system involves all of the challenges that crossover design has, with perhaps the only exception that you are involved in line level vs. speaker level settings. This is why I say it is far too much for the average audiophile, and best left to ARC systems like built into JL Audio subs or top notch digital preamps or integrateds like from Anthem.

Even if you do all the filtering right, the matter of the room remains.  To paraphrase the Bard:

Aye, and there is the rub, for in that room what sounds may come must give us pause and make cowards of us all.

Best,

Erik