The cathodyne does have markedly different output impedances at its two ports. It also has identical output levels if the load impedances are the same. These two statements do not conflict!
Think about building one with a valve which has high gm (so low output impedance as a cathode follower) and also high mu (so high anode impedance, made even higher in this case by the large cathode resistor). Then the cathode output approximates to a voltage source i.e. low impedance. The anode output approximates to a current source (i.e. high impedance) which happens to track the cathode current. As the currents are the same, identical loads will develop identical voltages.
If this is too complicated, think of it in another way. Consider two perfect voltage sources, one develops 1.1V, the other develops 101V. The first one has a series resistance of 100ohms, the second a series resistance of 100Kohms. So the external circuit sees an output impedance of 100 at the first, and 100K at the second. Now load each with 1K - you get 1V output from both!
The cathodyne remains balanced provided the external loads are identical. An extra resistor at the cathode helps maintain balance even when the loads do differ e.g. stray capacitance. This helps maintain overall balance, and reduces the risk of nasty things like unstable feedback loops on peak signals.
I had a quick look at the Valve Wizard website, and it looks good to me. I may put a link to it from my site.