I see signal imbalance, and if you're getting square out of one and sine from the other with the same input, something is definitely wrong. If you're inputting square and getting sine out, thats also telling me something is wrong.
Speaker phase might cause odd results if it switches it at the speaker output. With a common ground you'll likely be shorting the output when its in one position so the output will be very low and/or distorted. I'd probably run the scope - lead to chassis and use the + leads on the 8 ohm taps, bypassing the phase reverse stuff entirely just to rule out issues from that circuit.