Do water swirls or vortexes really spin in opposite directions depending on which hemisphere they are in?

I am sorry to let you know, but...this is definitely not true.  This is one of the most common "urban myths" in all of science, but not an accurate one.

The direction water takes going down a drain has to do with the specifics of how the water is injected into the basin, the shape and roughness of the basin and factors like that, but nothing to do with the rotation of the earth.

There IS a real effect due to the rotation of the Earth called the Coriolis Effect, but this is exhibited only over very large distance scales.  The Coriolis effect causes winds to deflect to the right of their motion in the Northern Hemisphere (left in the Southern), and causes ocean current to deflect also.  However, the Coriolis effect does not show up in your bathtub or toilet...

No it's not true. I was kind of disappointed when I found out this wasn't true, I thought that was neat. It turns out than any minor force action on the water can change which direction it swirls.