We use both photometric and spectroscopic data from the Hubble Space Telescope to explore the relationships among 4000 Å break (D4000) strength, colors, stellar masses, and morphology, in a sample of 352 galaxies with log(M */M ⊙) > 9.44 at 0.6 ≲ z ≲ 1.2. We have identified authentically quiescent galaxies in the UVJ diagram based on their D4000 strengths. This spectroscopic identification is in good agreement with their photometrically-derived specific star formation rates (sSFR). Morphologically, most (that is, 66 out of 68 galaxies, ~ 97 %) of these newly identified quiescent galaxies have a prominent bulge component. However, not all of the bulge-dominated galaxies are quenched. We found that bulge-dominated galaxies show positive correlations among the D4000 strength, stellar mass, and the Sérsic index, while late-type disks do not show such strong positive correlations. Also, bulge-dominated galaxies are clearly separated into two main groups in the parameter space of sSFR vs. stellar mass and stellar surface density within the effective radius, Σe, while late-type disks and irregulars only show high sSFR. This split is directly linked to the 'blue cloud' and the 'red sequence' populations, and correlates with the associated central compactness indicated by Σe. While star-forming massive late-type disks and irregulars (with D4000 < 1.5 and log(M */M ⊙) ≳ 10.5) span a stellar mass range comparable to bulge-dominated galaxies, most have systematically lower Σe ≲ 109 M ⊙kpc-2. This suggests that the presence of a bulge is a necessary but not sufficient requirement for quenching at intermediate redshifts.