The logic is simple: particularly, in a phono stage, where millivolt signals are amplified, we do not want any interaction between the triodes in a tube envelope. With all the triodes in an envelope seeing the same input signal, the interaction is minimized.
The calculation of the equalization network's values requires taking into account the output impedance of the first stage and the input resistance and capacitance of the second gain stage. Assuming a bypassed cathode resistor, the output impedance is easily derived from taking the effective rp of the input tubes in parallel and placing this value in parallel with the plate resistor's value, which is added to the 63.6k resistor's value. The input capacitance, i.e. the Miller-effect capacitance, is derived from taking the cathode-to-plate capacitance and multiplying it by the gain of the second grounded-cathode amplifier and then adding this sum to the grid-to-cathode capacitance. This capacitance adds to the .01 µF capacitor's value, which brings the equalization network into the correct alignment.
Of course, a two-gain stage, passive equalization layout is not limited to grounded-cathode amplifiers. In fact, as long as there are two gain stages bridged by a passive equalization network, just about any tube amplifier circuit can be used: the SRPP, the cascode, the grounded-grid amplifier (with some tweaking), and the common-cathode amplifier. Other circuits that could be used are the constant-current-draw amplifier, the plate follower, and the two tube feedback pair.