We only ever really measure the first two effective spin parameters $(\chi_\mathrm{eff}, \chi_p)$ — if at all. Our parameter estimation pipelines use a sampling prior flat in the component spins $(\chi_1, \chi_2, \cos\theta_1, \cos\theta_2)$. If I inject something where I know even the component spins exactly, my measurement garbles that up into something weird. Here's a toy model to understand that garbling.

For every $\chi_\mathrm{eff}, \chi_p, q$ that you specify, we do PE over the component spins using this crude approximate likelihood: $$\log \mathcal{L} = -\frac{\rho^2}{2}\!\left[ w_\mathrm{eff}^2\,\Delta\chi_\mathrm{eff}^2 + w_p^2\,\Delta\chi_p^2 + w_q^2\,\Delta q^2 + 2\rho_{eq}\,w_\mathrm{eff}\,w_q\,\Delta\chi_\mathrm{eff}\,\Delta q \right]$$ This is Gaussians in the effective-parameter space, mapped back through the prior on component spins.