The earth spins left to right. Why does the weather go from Left-Right - n.hemisphier, opposite in southern?

Outside of the tropics (more than about 23° from the equator) in both hemispheres, the dominant upper level flow is from the west. The weather doesn't go the opposite way in the southern hemisphere outside of the tropics.

Look at these satellite loops--first the southern, then the northern hemisphere...

http://vesta.cira.colostate.edu/RTSEC84_0.html

http://hadar.cira.colostate.edu/ramsdis/online/f6_...

Because of a weather process known as the Coriolis effect - due to the planet's rotation all air moves away from the equator in opposite directions. As the planet spins, that air is deflected through the various layers of atmosphere and rotates independantly due to various atmospheric influences.