  In this paper, we give a complete picture of Howe correspondence for the
setting ($O(E, b), Pin(E, b), \Pi$), where $O(E, b)$ is an orthogonal group
(real or complex), $Pin(E, b)$ is the two-fold Pin-covering of $O(E, b)$, and
$\Pi$ is the spinorial representation of $Pin(E, b)$. More precisely, for a
dual pair ($G, G'$) in $O(E, b)$, we determine explicitly the nature of its
preimages $(\tilde{G}, \tilde{G'})$ in $Pin(E, b)$, and prove that apart from
some exceptions, $(\tilde{G}, \tilde{G'})$ is always a dual pair in $Pin(E,
b)$; then we establish the Howe correspondence for $\Pi$ with respect to
$(\tilde{G}, \tilde{G'})$.
