These are Orbees, very special dog toys made by the excellent Planet Dog.
We call Orbees "Planets". Tala loves her Planet more than anything on the planet - except squirrels. Now that Tala can't chase squirrels, she likes to have the Planet with her at all times. When Diego runs off to chase squirrels, Tala grabs her Planet, and runs in little circles in her pen, squishing the ball in her jaws over and over.
Diego loves to steal Tala's Planet. He works at it, tearing off the continents one at a time. We would say, "Diego is destroying the Planet!"
To keep the peace, we ordered Diego his own Planet, and got a second one for Tala, too. I don't know what it is about these Orbees, but our dogs are crazy for them.