  Gravitational wave observations of binary black hole mergers probe their
astrophysical origins via the binary spin, namely the spin magnitudes and
directions of each component black hole, together described by six degrees of
freedom. However, the emitted signals primarily depend on two effective spin
parameters that condense the spin degrees of freedom to those parallel and
those perpendicular to the orbital plane. Given this reduction in
dimensionality between the physically relevant problem and what is typically
measurable, we revisit the question of whether information about the component
spin magnitudes and directions can successfully be recovered via
gravitational-wave observations, or if we simply extrapolate information about
the distributions of effective spin parameters. To this end, we simulate three
astrophysical populations with the same underlying effective-spin distribution
but different spin magnitude and tilt distributions, on which we conduct full
individual-event and population-level parameter estimation. We find that
parameterized population models can indeed qualitatively distinguish between
populations with different spin magnitude and tilt distributions at current
sensitivity. However, it remains challenging to either accurately recover the
true distribution or to diagnose biases due to model misspecification. We
attribute the former to practical challenges of dealing with high-dimensional
posterior distributions, and the latter to the fact that each individual event
carries very little information about the full six spin degrees of freedom.
