South, East, North

How many places are there on Earth so that if you travel 1 mile South, followed by 1 mile East, followed by 1 mile North, you will get back where you started from?

Remark: You can assume Earth is a perfect globe.

The answer is infinitely many. Of course, the North Pole is one such place. However, if you start close to the South Pole, such that after traveling 1 mile South you land on a parallel with total length of 1/N miles, N-integer number, then when traveling East you will encompass this parallel exactly N times and later will get back to the starting place. These are all places with the property described above.