Each stable polynomial p on the bidisk has a natural pair of commuting truncated Toeplitz operators associated to it that encodes important information about the distinct sums of squares decompositions (or Agler decompositions) associated to the rational inner function with p as its denominator. We review this construction and use it as motivation for a discussion of recent work on a detailed local theory of boundary zeros of stable polynomials. Joint work with Bickel, Pascoe, Sola.