  We consider various integrable two-parameter deformations of the $AdS_3
\times S^3 \times T^4$ superstring with quantum group symmetry. Working on the
string worldsheet in light-cone gauge and to quadratic order in fermions, we
obtain their common massive tree-level two-body S matrix, which matches the
expansion of the conjectured exact $q$-deformed S matrix. We then analyze the
behavior of the exact S matrix under mirror transformation -- a double Wick
rotation on the worldsheet -- and find that it satisfies a mirror duality
relation analogous to the distinguished $q$-deformed $AdS_5 \times S^5$ S
matrix in the one parameter deformation limit. Finally, we show that the
fermionic $q$-deformed $AdS_5 \times S^5$ S matrix also satisfies such a
relation.
