  We generalize the $\eta$-pairing theory in Hubbard models to the ones with
spin-orbit coupling (SOC) and obtain the conditions under which the
$\eta$-pairing operator is an eigenoperator of the Hamiltonian. The $\eta$
pairing thus reveals an exact $SU(2)$ pseudospin symmetry in our spin-orbit
coupled Hubbard model, even though the $SU(2)$ spin symmetry is explicitly
broken by the SOC. In particular, these exact results can be applied to a
variety of Hubbard models with SOC on either bipartite or non-bipartite
lattices, whose noninteracting limit can be a Dirac semimetal, a Weyl
semimetal, a nodal-line semimetal, and a Chern insulator. The $\eta$ pairing
conditions also impose constraints on the band topology of these systems. We
then construct and focus on an interacting Dirac-semimetal model, which
exhibits an exact pseudospin symmetry with fine-tuned parameters. The stability
regions for the exact $\eta$-pairing ground states (with momentum $\bm{\pi}$ or
$\bm{0}$) and the exact charge-density-wave ground states are established.
Between these distinct symmetry-breaking phases, there exists an exactly
solvable multicritical line. In the end, we discuss possible experimental
realizations of our results.
