There is no ground wire on a balanced headphone cable. For interconnects, a balanced cable is three wire per channel (+, -, and ground). For headphone balanced cables, there is only two wires per channel. You should just reterminate the plug on your cable and be done.
Thanks so much! I appreciate your thoughts. So confusing. I was thinking of balanced interconnects
For the most part, the only reason to go balanced is for more power to the headphones. Some folks claim they hear "better separation" or other benefits to balanced. Let your ears be the judge. On most amps that have balanced and SE, the balanced side will output quite a bit more power, so don't let the louder volume fool you into thinking the sound is better.
Most senngrados (and grados) don't a lot of power to drive well, so balanced isn't often necessary.
That said, on some amps the Balanced output uses a different topology than the SE output, or the SE output isn't does well, so the Balanced might actually sound better.