Much of the inner Milky Way's (MW) global rotation and velocity dispersion patterns can be reproduced by models of secularly-evolved, bar-dominated bulges. More sophisticated constraints, including the higher moments of the line-of-sight velocity distributions (LOSVDs) and limits on the chemodynamical substructure, are critical for interpreting observations of the unresolved inner regions of extragalactic systems and for placing the MW in context with other galaxies. Here, we use SDSS-APOGEE data to develop these constraints, by presenting the first maps of the LOSVD skewness and kurtosis of metal-rich and metal-poor inner MW stars (divided at [Fe/H] = -0.4), and comparing the observed patterns to those that are seen both in N-body models and in extragalactic bars. Despite closely matching the mean velocity and dispersion, the models do not reproduce the observed LOSVD skewness patterns in different ways, which demonstrates that our understanding of the detailed orbital structure of the inner MW remains an important regime for improvement. We find evidence in the MW of the skewness-velocity correlation that is used as a diagnostic of extragalactic bar/bulges. This correlation appears in metal-rich stars only, providing further evidence for different evolutionary histories of chemically differentiated populations. We connect these skewness measurements to previous work on high-velocity "peaks" in the inner Galaxy, confirming the presence of that phenomenon, and we quantify the cylindrical rotation of the inner Galaxy, finding that the latitude-independent rotation vanishes outside of lon ~ 7 deg. Finally, we evaluate the MW data in light of select extragalactic bar diagnostics and discuss progress and challenges of using the MW as a resolved analog of unresolved stellar populations.