Binormal Toeplitz operators on the Hardy space

We characterize binormal Toeplitz operators with analytic, or, coanalytic symbol functions. Furthermore, for a large class of nonanalytic, noncoanalytic Toeplitz operators which include Toeplitz operators with trigonometric or rational symbols, we prove that those Toeplitz operators are binormal if and only if they are normal. Some of the historically important examples of Toeplitz operators in the paper show that our problem is subtle and the above result is sharp.