The moon analogy is not quite right and in fact, it's easy to prove that Earth's moon at least describes proper orbital motion around the sun.
During a year, the moon describes a circular path around the sun (well, it's more like a triskaidecagon with rounded corners... I'll come to that.)
The Earth's orbit is 365.25 days, and the moon's orbit around the earth is 27.32 days - so the moon makes 13.37 orbits around the Earth in a year.
The Earth's orbit around the sun is 400 times the radius of the moon's orbit around the Earth, and the Earth's speed around the the sun is 30 times the speed of the moon around the Earth... this means that the moon will never display negative motion with relation to the sun, ie it can't move backwards.
So instead of transcribing a series of loops in its orbit around the sun - which would involve backwards motion - the moon transcribes a rounded, thirteen-sided polygon around the sun.
Of course, the most obvious body that doesn't orbit the sun is the sun itself...