  When $\lambda$ is a partition, the specialized non-symmetric Macdonald
polynomial $E_{\lambda}(x;q;0)$ is symmetric and related to a modified
Hall--Littlewood polynomial. We show that whenever all parts of the integer
partition $\lambda$ is a multiple of $n$, the underlying set of fillings
exhibit the cyclic sieving phenomenon (CSP) under a cyclic shift of the
columns. The corresponding CSP polynomial is given by $E_{\lambda}(x;q;0)$. In
addition, we prove a refined cyclic sieving phenomenon where the content of the
fillings is fixed. This refinement is closely related to an earlier result by
B.~Rhoades.
  We also introduce a skew version of $E_{\lambda}(x;q;0)$. We show that these
are symmetric and Schur-positive via a variant of the
Robinson--Schenstedt--Knuth correspondence and we also describe crystal
raising- and lowering operators for the underlying fillings. Moreover, we show
that the skew specialized non-symmetric Macdonald polynomials are in some cases
vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur
expansion of a new family of LLT polynomials.
