I think that they really believed that they were doing the right thing, and by their response, the ground controller must be wrong! But their situational awareness was off and they were not going where they thought that they were. Probably 90 degrees off like was mentioned before. How’d it happen? I don’t know but this confusion can happen so easily and that’s exactly why there are two pilots, there’s no shame in saying “Stop! I think we’re going the wrong way” or whatever the problem is even if you’re incorrect in the end.
As a recent example (I’m with a Part 91/135), on an arrival we briefed the approach that we were expecting from ATIS and then reviewed our expected taxi route as we always do. Then, while busy on the arrival, Approach changes the runway direction and switches everybody. So no big deal, you’re landing from the other side from what you previously briefed. But you are busy, recalculating landing numbers, briefing the new approach, loading it, etc, then get vectors, intercept and land, but you never briefed the new taxi route and now you’re on the ground. Everything still seems ok but when you turn off the runway something starts to seem off, you hesitate for a second as your brain says, “Wait a minute, I’m going the wrong way…” because you (from the prior briefing in the air) expected to turn the other direction. So you clear the runway, get confusing taxi instructions and have to stop to look at your taxi diagram to see what’s up before proceeding.
I don’t know what happened in their heads but the human brain just seems to have a hard time letting go of bad information. All it takes is a long day, weather challenges, an unexpected change or a little distraction and you are on your way to a mistake, hopefully not a dangerous/life threatening/career ending one. They approached the crossing runway, looked out the window (and maybe even saw the other airplanes lights!) but whatever, it’s not all processing yet, we are where we are supposed to be, clear left/clear right, cleared to cross…
This flying thing is really easy and fun until something goes wrong and you get on the wrong side of being ahead of the game. I’m so glad this wasn’t a huge disaster and everyone was able to go home. I bet the crew is just beating themselves up for crossing that runway.