  Various scenarios of cosmic inflation enhance the amplitude of the stochastic
gravitational wave background (SGWB) at frequencies detectable by the LISA
detector. We develop tools for a template-based analysis of the SGWB and
introduce a template databank to describe well-motivated signals from
inflation, prototype their template-based searches, and forecast their
reconstruction with LISA. Specifically, we classify seven templates based on
their signal frequency shape, and we identify representative fundamental
physics models leading to them. By running a template-based analysis, we
forecast the accuracy with which LISA can reconstruct the template parameters
of representative benchmark signals, with and without galactic and
extragalactic foregrounds. We identify the parameter regions that can be probed
by LISA within each template. Finally, we investigate how our signal
reconstructions shed light on fundamental physics models of inflation: we
discuss their impact for measurements of \emph{e.g.,} ~the couplings of
inflationary axions to gauge fields; the graviton mass during inflation; the
fluctuation seeds of primordial black holes; the consequences of excited states
during inflation, and the presence of small-scale spectral features.
