We consider the maximally symmetric two-Higgs doublet model (MS-2HDM) in which the so-called Standard Model (SM) alignment can be naturally realized as a consequence of an accidental SO(5) symmetry in the Higgs sector. This symmetry is broken (i) explicitly by renormalization-group (RG) effects and (ii) softly by the bilinear scalar mass term m122. We find that in the MS-2HDM all quartic couplings can unify at large RG scales μX∼1011-1020 GeV. In particular, we show that quartic coupling unification can take place in two different conformally invariant points, where all quartic couplings vanish. We perform a vacuum stability analysis of the model in order to ensure that the electroweak vacuum is sufficiently long-lived. The MS-2HDM is a minimal and very predictive extension of the SM governed by only three additional parameters: the unification scale μX, the charged Higgs mass Mh± (or m122), and tanβ, which allow one to determine the entire Higgs sector of the model. In terms of these input parameters, we present illustrative predictions of misalignment for the SM-like Higgs-boson couplings to the W± and Z bosons and, for the first time to our knowledge, to the top and bottom quarks.