  Given a locally compact abelian group $G$ and a closed subgroup $\Lambda$ in
$G\times\widehat{G}$, Rieffel associated to $\Lambda$ a Hilbert $C^*$-module
$\mathcal{E}$, known as a Heisenberg module. He proved that $\mathcal{E}$ is an
equivalence bimodule between the twisted group $C^*$-algebra
$C^*(\Lambda,\textsf{c})$ and $C^*(\Lambda^\circ,\bar{\textsf{c}})$, where
$\Lambda^{\circ}$ denotes the adjoint subgroup of $\Lambda$. Our main goal is
to study Heisenberg modules using tools from time-frequency analysis and
pointing out that Heisenberg modules provide the natural setting of the duality
theory of Gabor systems. More concretely, we show that the Feichtinger algebra
${\textbf{S}}_{0}(G)$ is an equivalence bimodule between the Banach subalgebras
${\textbf{S}}_{0}(\Lambda,\textsf{c})$ and
${\textbf{S}}_{0}(\Lambda^{\circ},\bar{\textsf{c}})$ of
$C^*(\Lambda,\textsf{c})$ and $C^*(\Lambda^\circ,\bar{\textsf{c}})$,
respectively. Further, we prove that ${\textbf{S}}_{0}(G)$ is finitely
generated and projective exactly for co-compact closed subgroups $\Lambda$. In
this case the generators $g_1,\ldots,g_n$ of the left
${\textbf{S}}_{0}(\Lambda)$-module ${\textbf{S}}_{0}(G)$ are the Gabor atoms of
a multi-window Gabor frame for $L^2(G)$. We prove that this is equivalent to
$g_1,\ldots,g_n$ being a Gabor super frame for the closed subspace generated by
the Gabor system for $\Lambda^{\circ}$. This duality principle is of
independent interest and is also studied for infinitely many Gabor atoms. We
also show that for any non-rational lattice $\Lambda$ in $\mathbb{R}^{2m}$ with
volume ${s}(\Lambda)<1$ there exists a Gabor frame generated by a single atom
in ${\textbf{S}}_{0}(\mathbb{R}^m)$.
