Critical subwoofer tip

Delay is to achieve the same impulse arrival time at the listening position. 10ms is about 11.25 feet. Delays around 1ms are more common.

An All-Pass Phase control does not affect all frequencies equally. It is used to get the best integration between the non-linear sub and main phase responses. see  PhaseControl.png (868×506) (ielogical.com) for an example

...and then there's the simple solution of delay for time resolution of waveforms.  Phase matching is easy....

I recently purchased a sub to augment the bottom end of my Spatial M3 open baffle speakers. Due to domestic constraints, one sub is all I will have in the living room. 
The sub is front firing, and placed about 18” from the rear of the Spatial. If you follow me here, I was concerned about the back wave (open baffle) cancelling out the sub’s forward wave. I made all my adjustments- xover freq, volume, Q, listening mode (Eq).
Things sounded very good- then I flipped the phase to 180 degrees.
This caused things to snap further into focus- I imagine the back wave of the OB speaker was interacting with the sub at some frequency, causing cancellation perhaps?
I mention this because it may very well be that some sub/ speaker interactions will not be affected by the switching of phase- sealed box speakers maybe, front ported, etc., on and on. Maybe rear ports with subs behind the speaker are affected differently.
This could explain why some people hear no difference when flipping phase- I have a separate home theater system, and for the life of me I never heard a difference when doing so. In this 2.1 channel system, it is readily apparent to me.
As I write, I also realize that the 2.1 system uses a sealed sub- the HT subs were ported. Maybe porting changes things too. Just more to consider...

More than a few post confuse polarity with phase. If nothing else, maybe this discussion will point this out.

180 degree phase shift LOOKS like reverses polarity, when graphed. However, 180 degrees phase shifted is actually 1 cycle delayed.

Inverted polarity will substantially reduce LF. (Subs out of polarity with the main speakers) 

Now, after the polarity is correct, we can optimize the phase. 

So, we must optimize sub: level, crossover, polarity and phase. 

I think it is safe to say that I am one of the confused ones. If we ignore changes in amplitude over time (impulses) and just stick with a sustained tone at a given frequency, how are two wave fronts 180 degrees out of phase not the same as two with the polarities reversed?
If I had a sub with speaker level inputs and no phase control, I would expect swapping the speaker leads to produce the same effect as flipping the phase 180 degrees.
BTW, I am enjoying these posts and appreciate the OP for starting it and everyone who has contributed. It has caused me to think more about things I "knew", or should have known, and how poorly I have considered some of those things in practice.