2006, v6, 140k.

I can select drive and 4th gear, but...
The gear selector cannot be physically moved to 3rd gear.
It just won't budge. It can be moved into drive, and then 4th, D, N, R, and P, but no other positions.

I feel like I've encountered this before and that there is a "secret handshake"... but I cannot recall/don't know if I'm misremembering.

Any thoughts?

Oh boy. This is a common problem. There is a pin that rides in a track as you move the shifter through the different positions. The track it rides in is plastic and wears down and develops a ridge that the pin can't get past. Do a search for shifter on here and also one for floppy shifter. There are several threads on here. The issue you now have may change to a floppy shifter if you try to force it where it doesn't want to go. The housing that the pin fits in will break off. The easiest fix, also the most expensive is to just go ahead and replace the whole shifter with a new one. I wouldn't recommend getting a used one. I tried that and the one I got was as bad as mine. They can be repaired if you're really handy. I fixed mine by removing the spring that applies pressure on the pin that rides in the track. The I cut the spring, thereby shortening it and greatly reducing the pressure on it. It works fine but it actually moves a little too easy from position to position. Anyway, you should be able to find a lot of info from the threads that are on here. Good luck