When I was young, swap the channels (ic, speaker cables, crystal gadgets, whatever)  used to be the first-step trouble shoot on problems like that. At least it would have vindicated the DAC immediately.

THIS.

Except...many people think "running the cables into the wrong channels" might hurt something. I seriously think people are getting stupider.

What kind are you looking for?  I am impressed with a recent pair of Canare GS6 wire with Canare RCAs for an analog out.  Made in Minneapolis and $24 for a meter.

Nothing in particular at the moment but just a general go-to for audio and data cables.

For interconnects the guys at proaudiola have yet to let me down.  Your choice of Mogami/Canare etc... cable made to custom length with Neutrik or Amphenol ends.  No nonsense goodness.

Sorry for the quasi-ad post but I think they provide a good service and they ship worldwide for reasonable cost.

many people think "running the cables into the wrong channels" might hurt something.

Of course, it is only for testing purposes. In the long term, the left channel, played through cables with red connectors, is not going to sound right at all. 

Any recommendations for cheap cables that work as they should?

Seconding Blue Jeans. I actually don't consider their cables to be "cheap!" But they come without bullshit.

Whilst I have actually bought some of their cables, I mostly use their articles as what's-actually-reasonable reference.

the left channel, played through cables with red connectors, is not going to sound right at all.

 

Come to think of it, left channels aren't supposed to sound right.