= Solution
The continuous internal <symmetry> is a common <rotation> around $z$:
$$
S_j^\pm\longmapsto e^{\pm i\theta}S_j^\pm,\qquad S_j^z\longmapsto S_j^z.
$$
Both the transverse exchange and the <Dzyaloshinskii-Moriya interaction> contain one raising and one lowering <spin operator>, so the phases cancel. Equivalently, $[H,\sum_jS_j^z]=0$ and the continuous group is <U(1)>.
The discrete internal <symmetry> is <time reversal>, $\mathcal T\mathbf S_j\mathcal T^{-1}=-\mathbf S_j$, together with $\mathcal T i\mathcal T^{-1}=-i$. Each term in $H$ contains two <spin operators>, so it is invariant. The operation is <antiunitary>; treating the reversal of all three <spin> components as a unitary <rotation> would violate the <spin commutation relations>. It reverses the <magnetization> and interchanges the two easy-axis ordered orientations. A convenient equivalent representative, composed with a $z$ <rotation>, is $R_z(\pi)\mathcal T$, which reverses $S^z$ while preserving $S^x,S^y$ as operators.
The statement about one continuous and one discrete internal <symmetry> describes generic parameters. At $D=0$, additional unitary $\pi$ <rotations> about transverse axes are allowed; at $D=u=0$, the continuous internal <symmetry> enlarges to full <spin> <rotation> symmetry. Spatial translations of the uniform chain are separate <symmetries> of the sites.
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