Starting points for a south-east-north walk loop

On a perfectly spherical Earth, from how many starting points can you walk $1$ km due south, then $1$ km due east, then $1$ km due north, and arrive back where you began?

Infinitely many. The North Pole works. Also, near the South Pole there is a latitude circle of circumference $1/k$ km for each $k=1,2,3,\dots$; start anywhere on the circle $1$ km north of one of these, and the eastward kilometre loops around $k$ times before you walk back north to the start.