If a plane was to fly from the middle of the USA to the middle of Australia would it fly left or right around the globe?

In college we had a spherical geometry question on a final exam. It went something like that: If a trip is flown from Phoenix to Melbourne, calculate the Great Circle departure heading from Phoenix and arrival heading at Melbourne. As I recall, the departure heading from Phoenix was a little south of south-west.

Here is a route taken by Qantas flight 8 from Dallas/Fort Worth to Sydney:

http://flightaware.com/live/flight/QFA8/history/20170124/0205Z/KDFW/YSSY

While this route is not exactly Great Circle, it is close enough, also Dallas is further east than Phoenix so the initial heading would be somewhat more southerly than that from Phoenix.

West, since you'd be using the earth's rotation to your advantage.

But you also have to consider jet streams. Jet streams are high altitude winds, and depending on your latitude, they either blow west to east or east to west. Since you'd be crossing several zones of jet streams, it doesn't matter which way you go. Sooner or later, you'll be fighting the wind. So the only factor is the earth's rotation.

It goes west, to California and then down from there.

It would fly westbound.

Aussie to Calif. will go right likely.

You would most likely find yourself flying from mid-American to the west coast and then west to Australia.

Navigation of airplanes is not described as "left" or "right" around the globe -
Airplanes fly almost straight towards their destination -
From the United States to Australia, it would be an approximate "true track" of 230º -

The best way for you to visualize - pull a string between departure and arrival on a school globe -
Hold one end by Los Angeles, and the other end on Brisbane -