  We provide a description of certain invariance properties of completely
bounded bimodule maps in terms of their symbols. If $\mathbb{G}$ is a locally
compact quantum group, we characterise the completely bounded
$L^{\infty}(\mathbb{G})'$-bimodule maps that send $C_0(\hat{\mathbb{G}})$ into
$L^{\infty}(\hat{\mathbb{G}})$ in terms of the properties of the corresponding
elements of the normal Haagerup tensor product $L^{\infty}(\mathbb{G})
\otimes_{\sigma{\rm h}} L^{\infty}(\mathbb{G})$. As a consequence, we obtain an
intrinsic characterisation of the normal completely bounded
$L^{\infty}(\mathbb{G})'$-bimodule maps that leave
$L^{\infty}(\hat{\mathbb{G}})$ invariant, extending and unifying results,
formulated in the current literature separately for the commutative and the
co-commutative cases.
