Something is goofy with my steering in that it makes a clunk when shifting the steering wheel from right to left. It won't do it if I'm just turning one direction, but the noise is quite pronounced when I move the wheel to the right, then immediate reverse direction to the left.

Underneath I can see the tie rod twisting, pivoting on what they're attached to, which look like ball joints or some such thing.. It seems when the tie rod pivots, the point at which it stops twisting is where it makes that loud clunk.

What is wrong here? I should also note that there is a good amount of play in the steering. From center, I can move the wheel a couple inches in either direction before the steering actually responds.

 

The drag link(or center link) connects to the passenger side with a tie rod end.
The drivers side tie rod connects from that link to the drivers side spindle/upright with another tie rod end.

You can move one of the wheels without the other one moving?!?!?
Or you can move the steering wheel some before the front wheels move?

If the latter, look at the slip and rag joints in the steering shaft before the steering box.
If the former, replace the tie rod ends and get an alignment before one of the wheels comes free and you have an accident.