  We extend a quantized skew Howe duality result for Type $\mathbf{A}$ algebras
to orthogonal types via a seesaw. We develop an operator commutant version of
the First Fundamental Theorem of invariant theory for $U_q(\mathfrak{so}_n)$
using a double centralizer property inside a quantized Clifford algebra. We
obtain a multiplicity-free decomposition of tensor powers of the
$U_q(\mathfrak{so}_{2n})$ spin representation by explicitly computing joint
highest weights with respect to an action of $U_q(\mathfrak{so}_{2n}) \otimes
U_q'(\mathfrak{so}_m)$. Clifford algebras are an essential feature of our work:
they provide a unifying framework for classical and quantized skew Howe duality
results that can be extended to include orthogonal algebras of types
$\mathbf{BD}$.
