FWIW I often listen to "The Pacific" television show with my headphones. The audio is definitely not left-right reversed. I think the issue is with your cable company.

My experience with cable company customer service is that you have to keep trying until you happen upon someone who's intelligent enough to understand your problem AND sincere enough to want to correct it.

Try calling at different times. Try visiting their website and leaving them a text message. Try yelling and getting pissed off. Try being nice and gentile. Be persistent.

Or, see this as an opportunity to get creative. Use PeterN's adapters, or your idea of wiring up your own adapter. Include a DPDT switch so you can reverse the audio whenever you want with a flip of the switch.