Headroom of cathodyne phase inverter (in vacuum tube amplifier)
maybe it has something to do with cathode resistor drop?


It totally has something to do with the cathode resistor drop. The bias setting of -1 volt to -1.5 volt refers to the voltage between grid and cathode and not from grid to 0 volts. A cathodyne has a cathode resistor and, if the anode quiescent current is 1 mA then that cathode resistor also takes that current and raises the cathode to a much higher quiescent voltage (say 40 volts using a 40 kohm cathode resistor). This will mean that the actual bias point for the grid is  38.5 volts to 39 volts and now, with a 24 volt p-p signal on top there is no problem. 


  if the bias is set to something like (-)1-1.5V, then peak voltage of
  input signal has to be smaller than 1.5V is we want to avoid
  distorition, right?


No, you are not accounting for the fact that the cathode voltage, under quiescent conditions might be 30 to 50 volts above 0 volts and this is due to anode current and the presence of the cathode resistor.

Simulation showing ECC82 waveforms




Cathode quiescent voltage = 50 volts
Anode quiescent voltage = 120 volts
Grid self-biased (like the more recent JFET type bias)


48 volt p-p input signal: -



Grid voltage stays below cathode voltage always.

72 volts p-p input signal: -



Now we see a difference - the ECC82 enters saturation and the grid voltage rises 0.6 volts above the cathode. Anode bottom peak is showing signs of heavy saturation.