east or west

If you take two airport pairs that are pretty much on the opposite sides of the world, does the airline ever have the equal choice to go east or west to end up in the same place? Or maybe two airlines run the route, and one goes east and the other goes west?

Yes. For example AKL - LHR has multiple routes in both directions on various carriers - e.g. via LAX, DXB, HKG, SIN, take your pick.

Air NZ used to fly in both directions themselves, though IIRC via Asia is codeshares only now.

(it’s a long trip regardless of which way you go!)

It depends on access to runways for emergencies for one.
I would love to fly direct NYC to Sydney, however after Baja, there isn’t much in the way of islands to land at in an emergency. There was a requirement to fly within x amount of hours of an airport. They have relaxed it in the last few decades(I am not sure of the specifics).
Imagine having a pressurisation issue, having to go down to 10,000 feet then putter along at 200-300kt to the nearest airport (2 hours away at 500-600kt)

There are also winds, at FL350+ they can be significant.
Sydney to LAX is generally 12 hours one way and 14 the other.

Not to mention War zones.

I am sure that there are more reasons.

Very few aircraft fly to opposite sides of the globe in a single hope.