The purpose of this paper is to develop a theoretical solution that describes the compressible bidirectional vortex. Similar studies by the authors have extended the Taylor and Culick profiles to incorporate the effects of compressibility in porous channels and tubes. Our study is prompted by the need to better understand the flow behavior at high speed in swirl-driven thrust chambers in which a reversing cyclonic motion is established. Such chambers have the advantage of promoting mixing, efficiency, and internal wall cooling. This is accomplished by confining combustion to an inner vortex tube that remains separated from the chamber walls by an outer stream of swirling, low temperature oxidizer. Our closed-form analytical solution is based on steady, rotational, axisymmetric, compressible, and inviscid flow conditions. It is constructed using a Rayleigh-Janzen expansion in the injection Mach number. At the outset, the compressible axial and radial velocities are captured along with the mantle movement at various Mach numbers and vortex Reynolds numbers. In view of the underlying assumption of axisymmetry, all properties are held constant about the chamber axis. We find that, so long as this condition is maintained, the swirl velocity remains invariant in the tangential direction.