How do you alter the phase relationship between I and Q in a quadrature signal?
As you already know, the quadrature error, also called quadrature skew, describes how far off the actual angle between I and Q is from the ideal 90 degrees.
It is one component of modulation error ratio.

There seem to be lots of people talking about how to measure quadrature skew,
but few people talking about how to compensate.
The two major approaches to compensation seem to be:


During initial factory tuning, some human watches an o'scope measuring the skew while manually tweaking the analog parts, or
During normal operation, the receiver internally estimates the quadrature skew and compensates digitally.


Often hardware designers deliberately build "off-tuned recievers".
With such hardware, when the transmitter repeatedly transmits "the same" constellation point, the raw I/Q point sampled by the reciever -- rather than staying close to a single fixed location in the I/Q diagram as with a perfectly tuned direct-conversion receiver -- instead rotates in a circle.

Such a receiver somehow estimates (typically in a FPGA or in software) how far the constellation has been rotated by that analog hardware, and corrects for it by rotating the I/Q sample in the opposite direction.
It is possible to simultaneously estimate the phase skew between I and Q and correct for that at the same time.
(In effect,
if you know the skew is S and you measure raw x,q coordinate,
you can deskew and find the true x,y in perfectly orthogonal coordinates
with y = qc - xt,
where c is some approximation of 1/cos(S) and t is some approximation of tan(S); often c = 1 and t=S (in radians) is adequate).

Further reading:


"Blind Equalization of Receiver In-Phase/Quadrature Skew in the Presence of Nyquist Filtering"
a patent search for "quadrature skew" compensation turns up a few.
"Effects of physical layer impairments on OFDM systems"
"Compensation for In-Phase/Quadrature Imbalance in Coherent-Receiver Front End for Optical Quadrature Amplitude Modulation"
"Correcting Imperfections in IQ Modulators to Improve RF Signal Fidelity"
"Estimation and Compensation of Impairments caused by Quadrature Modulators and Demodulators"