Page 6 and 7 of the attached document, the figure 6C is a variable phase shifter (90 degrees). Do the shunt capacitors need to be tuned at the same capacitance value both or it can be different?
They need to be the same value to keep the impedances of the I and Q branches equal. Note that the shift changes with frequency so simple RC or LC delays are only suitable for a very narrow range of frequencies and ideally need tuning for exact 90 degrees at each frequency used.
But doesn't shift one path to positive phase, while the other shifts to negative phase.
...in a way that the phase difference between both paths is always 90°? In a wide range of frequency..
Actually the example attached (with Xc=XL=50ohm) shows that the phase response is not wide range.
And to keep the impedance constant you have to change not only the values of C but also of the value of L, which makes the circuit pretty hard to tune.