Eastward is harder. Sleep research puts adjustment at roughly one day per time zone crossed going east, versus about two-thirds of a day going west. Flying west asks your body to stay up later; flying east asks it to fall asleep earlier, which most people find much harder.
Eastward is harder, and by a fairly consistent margin. Sleep research puts the adjustment at roughly one day per time zone crossed travelling east, against about two-thirds of a day travelling west.
The reason is easier to feel than to read. Say you normally go to bed at 11pm and you fly east across 8 time zones. Your body wants to sleep at 3pm local, so you have to force yourself to stay awake. Fly west across the same 8 zones and your body wants to sleep at 7am local, so you just stay up a little longer than usual. Most people find "stay up later" far easier than "fall asleep earlier."
On a round-the-world trip you cross every time zone regardless of direction, so what actually changes is where the big jumps land. Westward puts the largest shift, the transpacific crossing, in the easy direction. Eastward puts the smaller transatlantic shift in the hard direction at the start, then brings you home westward.
For most travelers this is a factor rather than the deciding one. If jet lag hits you hard, westward has a real advantage at the start of the trip, when you would rather be exploring than sleeping through your first two days.