Motivated by our search for a representation-theoretic avatar of double Grothendieck polynomials G_w(x;y), we give a new formula for G_w(x;y) based on Magyar's orthodontia algorithm for diagrams. We obtain a similar formula for double Schubert polynomials S_w(x;y), and a curious positivity result: For vexillary permutations w, the polynomial x_1^n \dots x_n^n S_w(x_n^{-1}, \dots, x_1^{-1}; 1, \dots, 1) is a graded nonnegative sum of Lascoux polynomials. This is joint work with Avery St. Dizier.