The cathodyne — also called the split-load or concertina inverter — makes its two opposite-phase signals from a single triode with equal resistors above the plate and below the cathode. The plate output is inverted, the cathode output is not, and because the loads are equal the two swings are equal. It costs half a dual-triode and a handful of resistors, and it needs no feedback loop to stay balanced.
That economy is why the medium-power tweeds used it. A single-ended amp needs no splitter, and a big amp wants the gain and balance of a long-tailed pair; the cathodyne sits between, giving a push-pull amp its phase split for the price of one triode. The trade is real: it provides no voltage gain, so the driving stage has to supply the full swing, and its two output impedances differ, so balance drifts as the amp is driven hard.
Pushed, that imbalance is part of the character — the give and bloom of a cranked tweed comes partly from a splitter that stops behaving perfectly right when the output tubes do. See how these stages sit in the signal path in the guide to reading a schematic.