  The Event Horizon Telescope (EHT) is a millimeter very long baseline
interferometry (VLBI) array that has imaged the apparent shadows of the
supermassive black holes M87* and Sagittarius A*. Polarimetric data from these
observations contain a wealth of information on the black hole and accretion
flow properties. In this work, we develop polarimetric geometric modeling
methods for mm-VLBI data, focusing on approaches that fit data products with
differing degrees of invariance to broad classes of calibration errors. We
establish a fitting procedure using a polarimetric "m-ring" model to
approximate the image structure near a black hole. By fitting this model to
synthetic EHT data from general relativistic magnetohydrodynamic models, we
show that the linear and circular polarization structure can be successfully
approximated with relatively few model parameters. We then fit this model to
EHT observations of M87* taken in 2017. In total intensity and linear
polarization, the m-ring fits are consistent with previous results from imaging
methods. In circular polarization, the m-ring fits indicate the presence of
event-horizon-scale circular polarization structure, with a persistent dipolar
asymmetry and orientation across several days. The same structure was recovered
independently of observing band, used data products, and model assumptions.
Despite this broad agreement, imaging methods do not produce similarly
consistent results. Our circular polarization results, which imposed additional
assumptions on the source structure, should thus be interpreted with some
caution. Polarimetric geometric modeling provides a useful and powerful method
to constrain the properties of horizon-scale polarized emission, particularly
for sparse arrays like the EHT.
