  We study a viable connection between the circular-equatorial orbits and
reflection symmetry across the equatorial plane of a vacuum stationary
axis-symmetric spacetime in general relativity. The behavior of the circular
equatorial orbits in the direction perpendicular to the equatorial plane is
studied, and different outcomes in the presence and in the absence of the
reflection symmetry are discussed. We conclude that in the absence of the
equatorial reflection symmetry neither stable nor unstable circular orbit can
exist on the equatorial plane. Moreover, to address the observational aspects,
we provide two possible examples relating gravitational wave astronomy and the
thin accretion disk which can put constraints on the symmetry breaking
parameters.
