Horses will curl their upper lip and press it to the back of their nose. This is called flehmen. A horse does this when it detects an odor worthy of pressing into a sensitive olfactory discrimination area called the voneronasal organ, which is located in the nasal cavity. The flehmen response increases the flow of air through the nostrils, which brings the scent openings behind the incisors on the upper palette to the onerously organ. This is seen more in males than females, especially stallions.
Perfumes and lotions on human hands can cause horses to curl their upper lip. A stallion will make this face when he examines a mare’s urine to find out if she is in heat. The vomeronasal organ is orientated with the brain’s sexual behavior center, and is used to decipher the pheromones a mare emits.
Twist looks like he’s laughing or smiling. I prefer to think he’s smiling.
Thanks to Kit from KSRP for the photos.