The old adage, "Where the head goes the body will follow." should be, "Where the eyes go, the head goes, and the body will follow." The eyes control movement more than most people realize. Where the eyes are looking controls where the head turns, and where the head turns controls how the body moves.
Lowering the chin or looking downward causes a reflex reaction that relaxes the back extensor muscles. This is useful if you are performing a forward roll in Judo class when you want to round the back to make the rolling motion easier or when you are performing curl-ups, but it is not useful when you are performing squats or a jump kick. Conversely, lifting the chin or looking downward lifts the chest and tenses the back and hip extensors, while simultaneously relaxing the abdominal muscles. This is useful when performing the pulling motion of a chin up, but it is not useful when abdominal curls.
This link between eye and head movement is useful when punching. For example, when performing a cross-hook-cross combination, look and turn the head slightly downward to the left as you throw the right cross, then look and turn the head slightly downward to the right as you throw the left hook, and then look and turn the head slightly downward to the left as you throw the final right cross The faster the eye movement, the faster the head movement, and the faster the head movement, the faster the body movement, which will increase the power of each punch.