I was also confused regarding the one where screen and plate provide reverse phase to the power tubes.
Basically works off the principle that in a pentode, screen current and plate current are in anti-phase (in order to achieve 'constant current'). In a normal pentode, under signal conditions, the screen is held at a constant voltage and when the plate current dips down (when the plate voltage swing peaks), the screen current zooms up - and vice-versa (all other things being equal).
In the above PI example, the screen is supplied with an independent current source (through the screen load resistor). The screen is not bypassed in any way, and the tube current is supplied via the 'constant current source'* by way of the tail resistor. So when the plate current dips, there is an alternative pathway for the tube current, through the screen, so the screen current rises - relative to the plate current (and vice versa when the plate current falls). Having a separate load resistances for the plate and for the screen, results in a differential in plate and screen voltage swings. But as PRR noted, this is not a very efficient phase inverter. This is because of the totally different physical (and electrical) properties of the plate vs the screen electrodes. (Note the different load resistances: 220k for plate and 120k for screen). Therefore, this PI typology would tend to suit some types of pentodes (ones which have higher-screen-current-relative-to-plate-current - and more robust screen grids) better than others (but even then it wouldn't be 'perfect')
*Re: the tail, the 'substantial' (22k) tail resistor at the cathode mimics a constant current supply for the pentode (like the tail in an LTP does), and the bootstrapping of the grid load resistor in combination with the bypassing of the cathode (which prevents changes in cathode voltage), helps to maintain the constant current function.