I've wanted this for a while, in combo with a head-tracking cabin camera. That way the windscreen would tint in specific areas over the sun, oncoming headlights, and glare.

You can't use LCs, as they don't have a real "clear" mode- they always block a very significant amount of light. You'd have to use electrochromatic paint or similar, as some trains and airplanes do. It's expensive and a chipped windscreen would be awful. You also can't get great resolution. I still wish somebody would do it.

Wouldn't darkening just absorb more heat? How about turning white, or a reflective coating? (Maybe not something liquid crystals could do, but a car with mirror shades would look cool.)

Probably true that white or reflective coating would be better, but even with black there would be much less green house effect inside the car. The windows would heat up though and would be cooled by the air outside.

I believe the reduced interior heating comes from the windows absorbing heat instead of the interior surfaces. Those hotter windows may radiate some inside, but they'll radiate a lot outside that never got into the interior.

Smart film/tint/glass on a car window: https://www.youtube.com/watch?v=c7xEcI1oSVs