You are completely confusing the action of a balanced connection with a balanced design. As has been stated in this thread by Tekdredger and Stabie, a balanced connection only cancels what noise is picked up by the cable connecting two balanced components (these can be unbalanced designs but have a balanced receiver and transmitter on each end). The distortion of the other component(s) is unaffected except for the distortion actually generated by the transmitting and receiving electronics.

A design that is completely balanced internally will cancel all even order harmonic distoriton that is generated within that component only, and pass through the odd order ones. If you have a power amplifier that is balanced all the way through, the only distortion components that that amplifier produces that will be present on it's output will be the odd order ones.

Any distortion that is generated by other components upstream will pass through that power amplifier unaffected because obviously the balanced amplifier doesn't know what is music and what is distortion in material that is presented to it's input.

The balanced amplifier however will add only odd order harmonic distortion to the signal.

One of the major reasons that vacuum tube amplifiers sound so musical is the fact that the majority of what distortion they produce is even order. What odd order distortion that they produce is essentially non-existant beyond the 3rd order.

Designing a component that is balanced all the way through from input to output is a bad design decision. There is absolutely no advantage sonically or theoritically to this type of design. There is however the very real and very audible downside of this type of design which is a result of the cancellation of all of the even order harmonic distortion components that that amplifier produces.

If an amplifier must have balanced inputs, the least soncally degrading way to accomplish this is to design a standard unbalanced circuit and add balanced-to-unbalanced circuitry at the amplifiers input which can be bypassed if the user is only using an unbalanced connection.