According to experts I've read or watched, it's all related to first, the animal's sense of who he is in the household, and second, eye contact or lack of same.
Years ago, I read an article by a veterinarian, in which he said cat lovers make eye contact with other peoples' cats, because they like cats, and are not afraid of them. But those who fear or dislike cats do not make eye contact.
Cats perceive this as admission of low status, and proceed to "walk all over them." That is why cats settle down on laps of ailurophobes. They interpret lack of eye contact as deference. Not so with dogs.
According to Cesar Millan, "The Dog Whisperer," dogs interpret lack of eye contact as a signal that you are alpha (dominant). Milan looks straight ahead, folds arms over chest, and strides towards any dog. Dogs back away, which Milan says signals acceptance of him as alpha.
I have copied Milan's strategy three times with dogs whose owners had not trained them. Each time, the dog backed up, calmed down, and accepted me as alpha.
I've always admired owners who control their dogs. Unlike cats, dogs want to please, which makes them easy to train. Unfortunately, many owners do not see the need for proper training, so their dogs rule them. Since I discovered the TV program, "Dog Whisperer," and watched fascinated as Cesar Millan turned the most annoying, bothersome dogs into quiet, cooperative pets, I have been a convert.
He does it by being the "pack leader," and dogs willingly defer to him. He trains owners to be pack leaders. When he leaves, the owner is still the pack leader, and the dog is his pleasant, obedient companion.
Pack leader or alpha dog; it's all the same concept. Owners should lead. Dogs should obey. And obey they do, as long as owners don't forget how to be pack leaders. The fact is, owners need training, which makes it possible for them to control their dogs.
What about cats? Cats, like dogs, establish a hierarchy in each group with one alpha, to whom the others defer. However, cats couldn't care less about pleasing people. The closest anyone comes to being a Cat Whisperer is the veterinary behaviorist, who succeeds by working with cat nature, and employing behavior modification.
People who look directly at cats put them on guard. But not making eye contact brings cats to you. Once you live with a cat, you can look deeply into each other's eyes and still be friends. You both know kitty is boss, and since cats are tiny compared to some dogs, that's okay. We love being their slaves!