What day of the year does the sun rise farthest south?

In the northern hemisphere, the Summer Solstice is on June 21 (plus or minus one day depending on the cycle of leap years), and the Sun is as far north of the equator as it ever gets. In the Southern Hemisphere, June 21 is the winter solstice.

In the northern hemisphere, the winter solstice is on December 21 (this is the summer solstice "Down Under") when the Sun is as far south of the equator as it ever gets.