I'd just like to add, the "Eye" was more than his heraldry and even more than a reference to his Palantír.
The Eye is ofttimes used when Sauron (*) focuses his will and dominance -and possibly but not necessarily his gaze- on someone. The subject's vision is obscured and blurred by a red ring that looks like an eye, as Frodo sees while in Mordor, and looking in Galadriel's mirror. It is also occasionally mentioned in the Silmarillion, long before Sauron took up the whole Eye heraldry and obtained a Palantír.
(*) And also Melkor, it isn't unique to Sauron alone: see HOME 10: Morgoth's Ring. The Eye of Morgoth is mentioned.
[I]In the sea without lees standeth the Bird of Hermes.
[/I][I]When all his feathers be from him gone, He standeth still here as a stone.
Here is now both white and red, And all so the stone to quicken the dead[/I][I].
The Bird of Hermes is my name, Eating my wings to make me tame.[/I]