What does the actual path of air within a turbojet engine look like?
(Source) Image shows CFD simulation for a jet engine carried out by Northern Arizona University.

Pretty much as your illustration, but more turbulent and not as many complete turns.

Let's take one rotor disk at a time.

Each blade on each disk is like a wing. Air is rushing in at it. And it, is moving down on the air (or up from the other side). Giving the blade an angle of attack.

In a wing, air is pushed down.

So, one disk will push down on the air on one side, up the other, left and right top and bottom—the stator disk will straighten it out for the following rotor. And so on, i.e., a stepping swirl.

If the stator reversed the flow, it will be an immense angle of attack for the receiving rotor, i.e., the rotor will stall. So, a stepping swirl makes more sense.

While a propeller, an axial compressor, and a turbine are different in what they do, I don't see why the shape of the corkscrew of the propeller I asked about here—would change for any rotating disk carrying airfoil blades—



(Source)

—except in magnitude and thermodynamic properties.

To complement the earlier CFD image, here are two more:



(Original image) Three stage transonic compressor.



(Source) Analysis of Rotor-Stator-Interaction and Blade-to-Blade Measurements in a Two Stage Axial Flow Compressor