The liquid core of a canonical two-fluid coaxial atomizer has been characterized using synchrotron X-rays. The high energy photons allow for high-speed imaging of attenuation through the dense liquid-gas jet core, resolving the internal structures that include entrapped air bubbles and the formation of liquid ligaments and bags. When the gas-to-liquid momentum ratio increases, the liquid core transitions from an intact column, where primary break-up happens several liquid diameters downstream, to a hollow crown with a downstream span comparable to the liquid diameter that disintegrates by shedding ligaments from its rim. At high gas momentum ratios (limited by the sonic velocity at the gas nozzle exit), this crown suffers partial dewetting and, when angular momentum is added to the gas, it dewets on a large section of the liquid injection needle circumference. This partial crown exhibits azimuthal motions along the circumference, on timescales much longer than the relevant flow timescales. The crown attachment to the liquid needle presents a bi-stable nature. The dramatic changes of the liquid core morphology, as the gas momentum and swirl ratios vary, have a strong impact on the gas-liquid boundary layers, which control the liquid break-up mechanisms and the resulting spray characteristics, such as droplet size distributions and the droplet volume fraction across the spray.