  Images of supermassive black hole accretion flows contain features of both
curved spacetime and plasma structure. Inferring properties of the spacetime
from images requires modeling the plasma properties, and vice versa. The Event
Horizon Telescope Collaboration has imaged near-horizon millimeter emission
from both Messier 87* (M87*) and Sagittarius A* (Sgr A*) with
very-long-baseline interferometry (VLBI) and has found a preference for
magnetically arrested disk (MAD) accretion in each case. MAD accretion enables
spacetime measurements through future observations of the photon ring, the
image feature composed of near-orbiting photons. The ordered fields and
relatively weak Faraday rotation of MADs yield rotationally symmetric
polarization when viewed at modest inclination. In this letter, we utilize this
symmetry along with parallel transport symmetries to construct a gain-robust
interferometric quantity that detects the transition between the weakly lensed
accretion flow image and the strongly lensed photon ring. We predict a shift in
polarimetric phases on long baselines and demonstrate that the photon rings in
M87* and Sgr A* can be unambiguously detected {with sensitive, long-baseline
measurements. For M87* we find that photon ring detection in snapshot
observations requires $\sim1$ mJy sensitivity on $>15$ G$\lambda$ baselines at
230 GHz and above, which could be achieved with space-VLBI or higher-frequency
ground-based VLBI. For Sgr A*, we find that interstellar scattering inhibits
photon ring detectability at 230 GHz, but $\sim10$ mJy sensitivity on $>12$
G$\lambda$ baselines at 345 GHz is sufficient, which is accessible from the
ground. For both sources, these sensitivity requirements may be relaxed by
repeated observations and averaging.
