head on a swivel
To be constantly moving your head left and right so that you do not miss anything approaching you from a blind spot
Mandy had her head on a swivel as she rode her bicycle in downtown traffic in order to avoid getting hit by a car.
keep your head on a swivel
To be alert, or just be watching around you
Guy1: Cops is lurking, keep your head on a swivel
Guy2: Aight g, you too
Guy2: Aight g, you too