  Gravitational lensing offers unique opportunities to learn about the
astrophysical origin of distant sources, the abundance of intervening objects
acting as lenses, and gravity and cosmology in general. However, all this
information can only be retrieved as long as one can disentangle each effect
from the finite number of observables. In the geometric optics regime, typical
of electromagnetic radiation, when the wavelength of the lensed signal is small
compared to the size of the lens, there are invariance transformations that
change the mass of the lens and the source-lens configuration but leave the
observables unchanged. Neglecting this ``mass-sheet degeneracy'' can lead to
biased lens parameters or unrealistic low uncertainties, which could then
transfer to an incorrect cosmography study. This might be different for
gravitational waves as their long wavelengths can be comparable to the lens
size and lensing enters into the wave-optics limit. We explore the existence of
invariance transformations in the wave-optics regime of gravitational-wave
lensing, extending previous work and examining the implications for
astrophysical and cosmological studies. We study these invariance
transformations using three different methods of increasing level of
complexity: template mismatch, Fisher Matrix, and Bayesian parameter
estimation. We find that, for a sufficiently loud signal, the degeneracy is
partially broken and the lens and cosmological parameters, e.g. $H_0$, can be
retrieved independently and unbiased. In current ground-based detectors,
though, considering also population studies, a strong constraint on these
parameters seems quite remote and the prevailing degeneracy implies a larger
uncertainty in the lens model reconstruction. However, with better sensitivity
of the third-generation ground-based detectors, a meaningful constraint on
$H_0$ is possible to obtain.
