My 2008 RX-350 headlights both pitch up and swing around when the car is started, but when I'm driving I've only ever notice the drivers side light following the Right turns in the road.

Is the passenger side light supposed to track the Left turns as well?

 

Mine is doing the same.I thing passsenger side moves when on right turn.

 

In general, the passenger side moves much less than drivers side.

 

Turning left - the driver's side light turns left.
Turning right - the passenger's side light turns right.

PS: If you can see that both headlights are moving when you turn them ON, but can't see the movements while driving - try to polish your headlights first (there are a lot of restoration kits online).