Looks like what I've occasionally seen through unbalanced single-ended preamps / recorders. Sometimes when mics or other circuits are near overload, or when a battery is weak.. basically the same thing I suppose. Although some waveforms tend to be naturally somewhat asymmetrical at times, like with trumpets.
Might play around with phase rotation. But I suspect that phase rotation of L vs R won't do what you want since both L and R visually appear to have the same asymmetry. However, some FM radio broadcasters manipulate phase of the difference channel as a way of reducing crest factor so as to achieve increased RMS values through the transmitter without increased peak values. With that in mind, maybe view the waveforms as Sum/Difference rather than L/R and look to see if the asymmetry predominates one channel more than the other. If it does, phase rotation of one channel (sum or difference) with respect to the other may achieve what you want.
I wonder if there are any plugin tools specifically designed to manipulate one side of a waveform to better match (or intentionally mismatch) the other? In this case, making the compressed looking positive going side match or at least more closely follow the negative going side. Shouldn't be difficult to do digitally, and would make for a useful if rather esoteric digital tool.