  The estimation of radiative modes is a central problem in gravitational wave
theory, with essential applications in signal modeling and data analysis. This
problem is complicated by most astrophysically relevant systems' not having
modes that are analytically tractable. A ubiquitous workaround is to use not
modes, but multipole moments defined by spin weighted spherical harmonics.
However, spherical multipole moments are only related to the modes of systems
without angular momentum. As a result, they can obscure the underlying physics
of astrophysically relevant systems, such as binary black hole merger and
ringdown. In such cases, spacetime angular momentum means that radiative modes
are not spherical, but spheroidal in nature. Here, we work through various
problems related to spheroidal harmonics. We show for the first time that
spheroidal harmonics are not only capable of representing arbitrary
gravitational wave signals, but that they also possess a kind of orthogonality
not used before in general relativity theory. Along the way we present a new
class of spin weighted harmonic functions dubbed ``adjoint-spheroidal"
harmonics. These new functions may be used for the general estimation of
spheroidal multipole moments via complete bi-orthogonal decomposition (in the
angular domain). By construction, adjoint-spheroidal harmonics suppress
mode-mixing effects known to plague spherical harmonic decomposition; as a
result, they better approximate a system's true radiative modes. We discuss
potential applications of these results. Lastly, we briefly comment on the
challenges posed by the analogous problem with Teukolsky's radial functions
