What is the algorithm for recognizing the I and Q components of the signal at the input of the quadrature demodulator?

Question

Algorithms

How is I or Q extracted from the signal entering the mixer by multiplying it by the carrier signal? Indeed, the ADC of the I-component and the ADC of the Q-component, in fact, still receive the same signal of the sum of the quadrature and common-mode components. How then is a constellation built from this signal?