Purely geometrical arguments show that there exist classes of homospectral
inflationary cosmologies, i.e. different expansion histories producing the same
spectrum of comoving curvature perturbations. We develop a general algorithm to
reconstruct the potential of minimally-coupled single scalar fields from an
arbitrary expansion history. We apply it to homospectral expansion histories to
obtain the corresponding potentials, providing numerical and analytical
examples. The infinite class of homospectral potentials depends on two free
parameters, the initial energy scale and the initial value of the field,
showing that in general it is impossible to reconstruct a unique potential from
the curvature spectrum unless the initial energy scale and the field value are
fixed, for instance through observation of primordial gravitational waves.
