A quick google search revealed some known power supply issues with the Mark Levinson No 27 Power amp. (It's not my amp so I didn't read further). The first thing I would check is the current draw from each amplifier and again when the cables are swapped. Get a clamp-on ammeter and an AC Line Splitter (Sears 81066).
A few things to consider:
1. Thinner wires have more resistance per foot. The longer the length, the more resistance causing a proportional voltage drop. And more resistance means more heat.
2. As voltage decreases, current increases.
3. Conductors and devices heat up as current rises, (eg.:fuses melt when current ratings are exceeded).
For whatever reason, my guess is that your two power cables are not at all alike. Perhaps the conductors, insulators or connectors in the bad cable are causing a voltage drop at the amplifier. That Sears AC line splitter also has voltage test points making it possible to measure voltage at the plug while under load.