  In this paper we consider the automorphisms of the double affine Hecke
algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action
on the generators and on the parameters, notably a symmetry $t_4$ which sends
the Askey-Wilson parameters $(a,b,c,d)$ to $(a,b,qd^{-1},qc^{-1})$. We study
how these symmetries act on the basic representation and on the symmetric and
non-symmetric Askey-Wilson (AW) polynomials and functions. Interestingly $t_4$
maps AW polynomials to functions. In the second part of the paper we look for a
version of non-symmetric AW functions that behave well under these symmetries.
