"As a circle is invariant under rotation, if we choose to number the objects 1 to 28, I can always assume that I've chosen the object at position 1. "
Imagine this: Mr. Left drove his car 5 miles. Then he drove it 3 miles. How many miles did Mr. Left drive.
As his name is Mr. Left the number of miles he drives will how many miles he has left after subtraction. And as $5 - 3 = 2$ he will have drive $2$ miles.
That logic makes as much sense as yours does. What does a circle be invariant under rotation have to do with anything?
Suppose that in position 1 was a bag of dog turds, and around on the other places were diamond rings and gold bullion, Rolex Watches, a first edition of Finnegan' Wake, and other useful things. You start by picking a diamond ring. But the guy who set this up says, "No, you have to choose the bag of dog turds first". You ask why and he says "Because the circle is invariant under rotation."
You give him dirty look, and pick the diamond ring in position 4, a micky mantle rookie card in position 5, and the topkapi daggar in position 6.
So the says "4,5,6. That's really the same thing as 1,2,3. So you picked the bag of dog turds, a bag of Channukkah chocolate, and a tuna sandwich made with truffle oil."
Eventually, it's time to walk away from this lunatic.