SUMMARY
Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation ˜Hµ(x;q,t)=˜Hµ*(x;t,q)\widetilde{H}_\mu(\mathbf{x};q,t)=\widetilde{H}_{\mu^\ast}(\mathbf{x};t,q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q=0q=0) when µ\mu is a partition with at most three rows, and for the coefficients of the square-free monomials in x\mathbf{x} for all shapes µ\mu. We also provide a proof for the full relation in the case when µ\mu is a hook shape, and for all shapes at the specialization t=1t=1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.