Just a random idea I had a minute ago.
Meet a cat's eyes. The cat looks away.
What we think: "I have shown my dominance to this cat by staring it down."
What the cat thinks: "I have looked away first, thereby proving myself to be the bigger cat."
Meet a cat's eyes. Look away first.
What we think: "Whatever. It's just a dumb cat."
What the cat thinks: "I AM THE MASTER OF ALL!"
~~~~~
Meet a dog's eyes. The dog looks away.
What we think: "Ha, I'm the alpha dog!"
What the dog thinks: "Was that food being poured in my bowl? FOOD!"
Meet a dog's eyes. Look away first.
What we think it means: "Oops, now he'll think he's in charge!"
What the dog thinks: "Squirrel? Did you see a squirrel? Where's the squirrel?! SQUIRREL!"