The analysis of anomalies in transit light curves caused by stellar gravity darkening has become a crucial tool for measuring the spin-orbit angle, ψ, which is the degree of misalignment between the stellar spin axis and the planetary orbital axis. The method has primarily been used to find perpendicular planets with asymmetric transits crossing both the stellar poles and the equator. In this work, the method is used for the first time to explore the possibility of a flipped orbit, rather than searching for perpendicular planets with transit asymmetries. Since aligned or counter-orbiting planets with the transit chord passing through the equator will maintain a symmetric transit despite gravity darkening, knowledge of the sky-projected obliquity λ and rotational velocity can help rule out polar orbits and suggest flipped orbits, which have yet to be found. Applying the spectroscopic constraints of sky-projected obliquity λ=-179.7±3.8° and vsini* = 84.2 km s-1, the gravity darkening analysis of symmetric TESS transits of hot Jupiter KELT-19Ab reveals that the spin-orbit angle is larger than 125°. As such, KELT-19Ab is likely the first known planet to have a flipped orbit. Recent numerical simulations have shown that such counter-orbiting systems can arise from Kozai-Lidov oscillations induced on primordially misaligned planets.