It can (and does) go pretty much any way it wants to get the best winds, and the outward will often route markedly differently to the return.
Great Circle Singapore to New York is 9,530 nm, which passes close to the North Pole, thus due north from Singapore and due South into New York.
However if you go overhead London, say, the two Great Circles together come to 10,220 nm, only another 690 nm, which in certain weather and routes available combinations may give the shortest time.
If it goes the "other way round", over the North Pacific and overhead Anchorage, Alaska, that's only 10,038 nm. Curously, this routing is so much over land that I reckon you could do it without ETOPS
The jetstream does tend to blow from the west, and it is good to follow it where you can. The A340 demonstration flight "round the world", Toulouse-Auckland-Toulouse, did exactly the same thing, out over Asia, back over America.