The correct answers are:
Spring equinox - September 22-23
Summer solstice - December 21-22
Fall equinox - March 21-22
Winter solstice - June 20-21
The equinoxes and the solstices on the Southern Hemisphere are opposite than the ones on the Northern Hemisphere, and the main reason for that is the inclination if the Earth. While one hemisphere gets more sunlight in one period of the year, the other gets less, and vice versa. That affects the appearance of summer and winter, spring and autumn, which are in contrast on the two hemispheres.
When its summer on the Northern Hemisphere, its winter in the Southern Hemisphere, and the other way around, which affects the dates of the solstices and equinoxes.
