  A contrastive investigation of the symmetry aspects of phonons in monolayer
chromium trihalides and their Janus structures Y$_3$-Cr$_2$-X$_3$ (X, Y = F,
Cl, Br, I) by group theory is presented. We first classify all phonons at the
Brillouin-zone center ($\Gamma$) into the irreducible representation. Then the
infrared and Raman activity of optic phonons, Raman tensors, and the possible
polarization assignments of R active phonons are predicted. Base on these
results, we clarify the the discrepancy about the Raman activity o optic modes
in monolayer CrI$_3$. Besides, we find that the Raman and infrared spectra for
X$_3$-Cr$_2$-X$_3$ are exclusive, whereas that for Janus Y$_3$-Cr$_2$-X$_3$ are
coincident. This distinction is vital for optic spectra identification of Janus
Y$_3$-Cr$_2$-X$_3$ monolayer from X$_3$-Cr$_2$-X$_3$ monolayer. In addition, we
derive the symmetry-matched phonon eigenfunctions and corresponding schematic
representations of the eigenvectors for both F$_3$-Cr$_2$-I$_3$ and
I$_3$-Cr$_2$-I$_3$ monolayer, which demonstrate intuitively the origin of
phonon chirality and magnetism. At last, our analysis indicates that the
spin-phonon coupling, the magneto-optical effect of infrared and Raman active
phonons, and phonon chirality should be observed in Janus Y$_3$-Cr$_2$-X$_3$
monolayer as that and even easier than that in X$_3$-Cr$_2$-X$_3$ monolayer.
Our work provides a detailed guiding map for experimental characterization of
Y$_3$-Cr$_2$-X$_3$ monolayer, and also reveals important effects of optic
phonons in Janus Y$_3$-Cr$_2$-X$_3$ monolayer.
