  We present an in-depth analysis of a newly proposed correlation function in
visibility space, between the E and B modes of the linear polarization,
hereafter the EB-correlation, for a set of time-averaged GRMHD simulations
compared with the phase map from different semi-analytic models as well as the
Event Horizon Telescope (EHT) 2017 data for M87* source. We demonstrate that
the phase map of the time-averaged EB-correlation contains novel information
that might be linked to the BH spin, accretion state and the electron
temperature. A detailed comparison with a semi-analytic approach with different
azimuthal expansion modes shows that to recover the morphology of the
real/imaginary part of the correlation function and its phase, we require
higher orders of these azimuthal modes. To extract the phase features, we
propose to use the Zernike polynomial reconstruction developing an empirical
metric to break degeneracies between models with different BH spins that are
qualitatively similar. We use a set of different geometrical ring models with
various magnetic and velocity field morphologies and show that both the image
space and visibility based EB-correlation morphologies in MAD simulations can
be explained with simple fluid and magnetic field geometries as used in ring
models. SANEs by contrast are harder to model, demonstrating that the simple
fluid and magnetic field geometries of ring models are not sufficient to
describe them owing to higher Faraday Rotation depths. A qualitative comparison
with the EHT data demonstrates that some of the features in the phase of
EB-correlation might be well explained by the current models for BH spins as
well as electron temperatures, while others may require a larger theoretical
surveys.
