The square wave tells me a surprising amount about a headphone. I haven't completely verified all of these observations and I will admit that I am not sound engineer, but I've noticed a few patterns.
The amount the 30hz square wave is bowed inward roughly shows the amount of bass roll-off that is occurring and when combined with the THD+noise you can roughly determine the amount of roll off and when it starts after seeing enough of them for a dynamic headphone. If it is too far bowed in and goes below 0, the bass response tends to be heavily rolled off and poor. In the rare case that it isn't bowing inwards and instead pushes upwards there is an increasing amount of sound as the frequency gets lower -- you can see this clear as day in the XB500's 300hz square wave response when you compare it against the frequency response curve.
Square waves are also probably the only way I've ever seen of telling if a headphone is overdamped or not. If the 30hz square wave looks boxy, it tends to be coming from a planar magnetic or orthodynamic because dynamic headphones almost never show complete linearity at 30hz. The exception is when the headphone is slightly overdamped like the KRK KNS 6400 or the DT1350. Although the shape appears like the optimal boxy shape, there is a funky bump and no initial spike. The closeness to the optimal square wave and the staying above 0 the duration means the bass is still tight and strong but lacks the "full strength" that a non-overdamped headphone with great bass extension like the LCD-2 would have caused by the "slow" start of the sound that doesn't start with a huge peak. This can also give the appearance of lack of bass (hence why I think the KRK KNS 6400 is reviewed as having zero bass and why the reviews completely contradict the graphs.) You can definitely tell that some of the T50RP mods were overdamped a bit from their mass loading and damping mods to get the very nice bass response from the square waves. I suspect that Fischer-Audio's products are a little overdamped as well from the 30hz response of the FA-003 and the reviews that suggest that the FA-011 is overdamped.
One thing about great square waves is when a headphone reaches linearity quickly. Tyll's graphs for the HD800, LCD-2, HE-500, and K701 all show excellent speeds at reaching linearity on the 300hz. I think the quick resolution of the square wave to linearity is what implies a headphone's imaging ability. You can see how the HD800 resolves its 300hz wave slightly faster than the HD600 and HD650. My gut feeling is that speed to linearity implies detail while the flatness of the tailing end of the square wave implies imaging ASSUMING the frequency response is not so colored that it no longer sounds real, but I can't say for certain. The Shure SRH940's quick resolution is why I believe people see it as a very detailed headphone that some say can fight with the HD800. The DT-48E also resolves very quickly, which explains why so many rabid followers of the DT-48E brag about its detailing, but its funky 30hz resolution implies to me that it has a very overdamped bass response, which explains why people see it as so bass light, even though the measurements aren't rolling off the bass that hard and are pretty flat after the initial fall. You can also see why some people think that Ultrasones, even the highest end Ultrasone headphones, have difficulty with imaging from this -- they constantly have a funky, wavy 300hz response and a colored bass response.
I have found a few exceptions, of course. One is the ATH-M50, which resolves the 300hz wave very quickly, abnormally quick, really. It has one huge spike which then completely dies. I think that odd bit might be part of why it doesn't resolve as well as the more hi-fi headphones. Another issue is that these low frequency graphs don't tell me much of anything about the treble. Take the Superlux HD681, for example -- beautiful sine waves for the price, but it tells you nothing about the piercingly bright treble.
Now, just for an example of how I would read the graphs... I opened up this PDF using only the square waves. http://www.innerfidelity.com/images/BowersandWilkinsP5Test2.pdf
The bass response looks flat, but overdamped because of the boxy shape, but a lack of a leading spike and because it rolls into the square bass with a hump. This implies that the bass response will be there, but it will lack "body" and "strength" because it is overdamped. The 300hz square wave implies a rising amount of bass as the frequency gets lower (the rising slope) but it has a relatively quick resolution to linearity with only three real spikes. This probably means it has a bassy response, because not many non-colored headphones increase in loudness at the 300hz range. The flatness of the line implies to me that the overall detail quality is good. The overall imaging is an unknown because the response is colored, which could skew things.
As for the Mixmasters posted earlier, here's how I would read the squares. The bass response looks well extended -- the 30hz wave is not quite as filled out as the HD800, so the extension isn't quite -that- good, but it is good nonetheless It has the typical dynamic underdamped headphone bowed in shape. The 300hz resolves to linearity relatively quickly, but it continues to oscillate, implying some imaging issues. For a $250 closed headphone, they seem pretty in line with what other companies offer. It has better bass response than the SRH940, but I would expect the detailing and imaging to be a bit off with the ringing in the 300hz and the frequency response.
... This post got way too long very fast.