Summary: We introduce an action of a discrete subgroup \(\varGamma\) of \(\text{SL}(2,\mathbb{R})^n\) on the space of pseudodifferential operators of \(n\) variables, and construct a map from the space of Hilbert modular forms for \(\varGamma\) to the space of pseudodifferential operators invariant under such a \(\varGamma\)-action, which is a lifting of the symbol map of pseudodifferential operators. We also obtain a necessary and sufficient condition for a certain type of pseudodifferential operator to be \(\varGamma\)-invariant.