Reflection symmetries of q-bernoulli polynomials

A large part of the theory of classical Bernoulli polynomials B n(x)'s follows from their reflection symmetry around x = 1/2: B n(1 - x) = (-1)n Bn(x). This symmetry not only survives quantization but has two equivalent forms, classical and quantum, depending upon whether one reflects around 1/2 the classical x or quantum [x]q.