Sorry Mike, haven't been back here for a while. I could attach a parcel shapefile, but I can tell you it will match the landscape shapefile perfectly as it does below.. the landscape shapefile was built by copying the parcels in ArcMap so they would align, which they do.
When you add a shapefile to My Content and publish hosted features from it, all the vertices of the data are loaded into the database. However, clients can draw the data in a generalized form. For example, the JavaScript API, upon which the ArcGIS.com map viewer is built does generalize the data.
I'm experiencing a similar issue, although the polygons are from separate layers. I now use only hosted features to avoid the misalignments the triggered this thread, but there are still generalizations in the map viewer that cause previously aligned boundaries to not align perfectly. I have a parcel feature service with zoning and city limit feature services which should align perfectly, but instead they meander somewhat like your examples. And for some reason the far smaller datasets of zoning and city limit are much more generalized than my parcel features. Just don't get it.