The N-color quantum Ashkin-Teller spin chain is a prototypical model for the study of strong-randomness phenomena at first-order and continuous quantum phase transitions. In this paper, we first review the existing strong-disorder renormalization group approaches to the random quantum Ashkin-Teller chain in the weak-coupling as well as the strong-coupling regimes. We then introduce a novel general variable transformation that unifies the treatment of the strong-coupling regime. This allows us to determine the phase diagram for all color numbers N, and the critical behavior for all . In the case of two colors, N = 2, a partially ordered product phase separates the paramagnetic and ferromagnetic phases in the strong-coupling regime. This phase is absent for all , i.e., there is a direct phase boundary between the paramagnetic and ferromagnetic phases. In agreement with the quantum version of the Aizenman-Wehr theorem, all phase transitions are continuous, even if their clean counterparts are of first order. We also discuss the various critical and multicritical points. They are all of infinite-randomness type, but depending on the coupling strength, they belong to different universality classes. © 2015 The Royal Swedish Academy of Sciences.