= Solution
In the ordered product basis $(|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle)$, the <two-spin XY Hamiltonian> is
$$
\boxed{H=\begin{pmatrix}-2h&0&0&0\\0&0&-2J&0\\0&-2J&0&0\\0&0&0&2h\end{pmatrix}.}
$$
The transverse term exchanges opposite spins and leaves parallel spins unchanged. Diagonalizing its two-dimensional block gives the complete <orthonormal eigenbasis>
$$
\boxed{\begin{array}{c|c}
\text{state}&\text{energy}\\\hline
|\uparrow\uparrow\rangle&-2h\\
(|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle)/\sqrt2&-2J\\
(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt2&2J\\
|\downarrow\downarrow\rangle&2h
\end{array}}
$$
The antisymmetric state is the <singlet state>; the other three are <triplet states>. Accidental equality of <energies> does not invalidate this <eigenbasis>.
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