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.

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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

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