I asked an earlier question about Geocoding (I initially called it Georeferencing, but have since learned the difference) addresses from the Saint Louis Metropolitan Police Department (SLMPD) crime data. I wound up having to geocode using ArcGIS online to project to a basemap, and then downloaded the resulting projected layers to desktop (ArcMap). That part is fine.
The part that is NOT working is that when I try to add these layers to my other layers (projected at NAD_1927_StatePlane_Missouri_East_FIPS_2401), by projecting them from their source data (WGS_1984_Web_Mercator_Auxiliary_Sphere) to _1927, it doesn't work. It consistently projects it to the exact same location it has as _1984, which is slightly higher and to the left of _1927. I know this can work. I did it with an earlier section of data from SLMPD via the ArcGIS online method. The problem is I did not keep a record of how I fixed it, nor do I remember how I did it. I have tried projection to _1927, projection to _1983(US feet) and THEN projection to _1927, batch projection in the same manner, creating a custom GeoTransformation from _1984 to _1927 and then Projecting, and by defining projection to another source and then projecting to _1927. None of them have worked. I think I'm stuck because I'm fixating too much and can't see outside the box, so I thought I'd ask for suggestions from the GIS crowd.