Ded Moroz (aka Santa Claus) brought me the new equipment, so my lab contains following items now:
- soldering station
- Rigol DS1102E scope
- Victor VC2002 functional signal generator
- Mastech HY3003-3 DC power supply
- HP Compaq nc6000 laptop
- mountains of electronic components
- several convolutions of the brain, somewhat straighten arms and other physiological equipment

Lab setup
Electronic details will likely do not match my next scheme (frequent reader will obviously notice this essentially a rule), even although I have really lots of passive and (noticebly smaller amount) active components now. I got several rectifying bridges at least, so will not use part of the transistor as a diode, and couple of the stabiliers.
But we will see, and now let's return to my first LM3886 amplifier with differential PSU, which did not really work like expected. I've attached differential signal from the lab PSU and volia, 20+ amplified output signal. That's how it looked.

Medium frequency sine signal looks good. Well, it almost always looks good.

Low frequency square signal
It shows some problems, but apparently it is the generator, which is not able to produce correct square signal on the small frequencies. If you could see how it tries to do it at 1Hz and smaller frequencies. Looks like it does not use comparator at the output chain for the square signal and just relies on differential and integrating circuits of the operation amplifier, but I may be very wrong of course.

High frequency square signal
High frequency (about 15 kHz, not really high, but high enough for the sound amplifier) amplification produces some delay and stabilizing pike, but it is expected when working with interpolation active filters, which are likely used in LM3886.
Found interesting problems with my LM3886 amplifier and chip behaviour in general during my tests:
- Amplifier differently drains input voltage, when it is turned on, I get -10V and +20V on the terminals. This may be related to the fact, that ground point in the PSU is not really a ground, since it is not 'connected' to the input voltage source, i.e. looks like current does not flow between ground and positive or negative terminals (when connecting voltmeter for example).
- after some short enough time (about several minutes) amplifier stops producing output signal, I checked input voltage, and found that negative voltage dropped from -10V down to -8.5V. Experimenting with the input voltage I found that when negative input voltage drops below (in absolute numbers, i.e. -10 is 'below' than -20 here) -9V, LM3886 is closed and no output signal is produced.
I think input voltage flows likely because of some problems with bypass capacitors connection or wrong ground potential. That analysis I will keep for the next experiments.
Some new scheme will be implemented soon and likely be tested with the loudspeakers during the New Year vacations and celebrations. Stay tuned for the news (operation amplfiers, comparators, timers, stabilizers, bipolar (including Darlington pairs) and mosfet transistors, they all will be used for something I do not yet know exactly what :)!
I've not used the model of PSU which you have but from your description I think you are using it incorrectly.
The ground (green) terminals should be completely ignored. The DC supplies will then float with respect to ground. To make a -20 - 0 - +20V supply you should set both of the variable supplies to 20V and then connect the negative of one supply to the positive of the other. The voltage regulation within the PSU will then make the remaining positive terminal 20V above the connected terminals and the remaining negative terminal 20V below.
You should not set the supply for 40V and assume that the green terminal will end up midway between.
Documentation says that it is a case ground terminal and does not specify if output voltage can be measured relative to it, so I checked if it can be used as a ground for the differential signal. It can not.
Connecting PSUs in a series should fix this issue of course.
From picture it looks like you have used the oscilloscope in AC coupling mode, which is a likely explanation for the distortions you see... )
btw, nice speakers
abr
It should be AC mode, although not sure why AC coupling will produce square shifts. I thought of the active output filters in the chip, which work as interpolation polynomial and thus 'explodes' at the strong signal turns.