I've only had an Intro. to Astronomy course about 20 years ago, but I'm pretty sure the the way you've listed the equinox dates for the southern hemishere isn't quite right.

Google knows all:

Vernal Equinox: The sun is on the equator moving north, and marks the beginning of spring, (in the southern hemisphere this is the beginning of autumn).

I'd list it like this:

Northern Hemisphere:
Spring, March 21 (Vernal Equinox)
Summer, June 22 (Summer Solstice)
Autumn, September 22 (Autumnal Equinox)
Winter, December 22 (Winter Solstice)

Southern Hemisphere:
Autumn, March 21 (Vernal Equinox)
Winter, June 22 (Winter Solstice)
Spring, September 22 (Autumnal Equinox)
Summer, December 22 (Summer Solstice)

Cheers,

Brent