"Fingerprinting" traits of a horse

They are called swirls or whorls.

Swirls on horses are the equine equivalent of fingerprints on humans. With no two patterns alike, they are like stamps marking each individual’s unique identity.

In a number of breeds swirl patterns are used as identification for horses: the Arabian Horse Association requires a record of swirls as identification for racing, and the American Quarter Horse Association uses swirls as a means of identification for solid color horses. When Thoroughbreds are registered, part of the procedure is to record the swirl patterns on the face and on both sides of the neck.

A swirl, or whorl, as it is also called, can be defined as a distinctive pattern in the lay of hair on an animal, often having the design of whirling, flexible spokes rotating about a center.

_{Getting in TTouch with Your Horse: how to assess and influence personality, potential, and performance
by Linda Tellington-Jones}

There is even a name for the study of hair swirls or whorls in horses: swirlology or whorlology.

Further readings:

• http://foxpointfarm.com/Swirlology.html
  
• http://horsemanship-journal.com/theory/whorlology/