  The Event Horizon Telescope (EHT) has imaged two supermassive black holes,
Messier 87* (M87*) and Sagittarius A* (Sgr A*), using very-long-baseline
interferometry (VLBI). The theoretical analyses of each source suggest
magnetically arrested disk (MAD) accretion viewed at modest inclination. These
MADs exhibit rotationally symmetric polarization of synchrotron emission caused
by symmetries of their ordered magnetic fields. We leverage these symmetries to
study the detectability of the black hole photon ring, which imposes known
antisymmetries in polarization. In this letter, we propose a novel
observational strategy based on coherent baseline-averaging of polarization
ratios in a rotating basis to detect the photon ring with 345 GHz VLBI from the
Earth's surface. Using synthetic observations from a likely future EHT, we find
a reversal in polarimetric phases on long baselines that reveals the presence
of the Sgr\,A* photon ring in a MAD system at 345 GHz, a critical frequency for
lengthening baselines and overcoming interstellar scattering. We use our
synthetic data and analysis pipeline to estimate requirements for the EHT using
a new metric: ${\rm SNR}_{\rm PR}$, the signal-to-noise ratio of this
polarimetric reversal signal. We identify long, coherent integrations using
frequency phase transfer as a critical enabling technique for the detection of
the photon ring, and predict a ${\rm SNR}_{\rm PR} \sim 2-3$ detection using
proposed ngEHT parameters and currently-favored models for the Sgr A* accretion
flow. We find that higher sensitivity, rather than denser Fourier sampling, is
the most critical requirement for polarimetric detection of the photon ring.
