  The ring-like images of the two supermassive black holes captured by the
Event Horizon Telescope (EHT) provide powerful probes of the physics of
accretion flows at horizon scales. Specifically, the brightness asymmetry in
the images carries information about the angular velocity profile of the inner
accretion flow and the inclination of the observer, owing to the Doppler boosts
photons experience at their site of emission. In this paper, we develop a
method for quantifying the brightness asymmetry of black-hole images in the
Fourier domain, which can be measured directly from interferometric data. We
apply this method to current EHT data and find that the image of Sagittarius A*
(Sgr A*) has an unusually low degree of asymmetry that is even lower than that
inferred for M87. We then use a covariant semi-analytic model to obtain
constraints on the inclinations and velocity profiles of the inner accretion
flow for Sgr A*. We find that the lack of significant brightness asymmetry
forces the observer inclination to uncomfortably small values ($6-10^\circ$),
if the plasma velocity follows Keplerian profiles. Alternatively, larger
inclination angles can be accommodated if the plasma velocities are
significantly sub-Keplerian and the black hole is not spinning rapidly.
