Dean Hoffman wrote
Commander Kinsey wrote

Why do people say left hand down instead of left?
We all know which way to turn the wheel to make
the car go left, and it's the same in reverse. And if
I'm reversing looking over my shoulder, one hand
isn't on the wheel so the phrase makes no sens

The only thing I can think of is from the old days.

Dunno, hard to see that unless you mean say the military during WW2 etc.

There was a time when drivers used hand signals
to signal left, right, or stop with their left hands.

Nope, with their right hands, Britain is right hand
drive and that phrase doesnt appear to be that
common outside right hand drive countrys.

And there was no down involved. There was a way to
indicate a left turn by the driver in a right hand drive
car but I cant remember what what it was. Hand moving
downward indicated slowing down before lights were
common.

That might still be legal

Yeah, it is still used by bicycle riders.

but I don't remember anyone using them.

I did see it very rarely indeed in my early days
and still see it quite a bit with bicycle riders.