Actually, your initial understanding is inaccurate (or at least incomplete). GEOGRAPHY stores LONGITUDE/LATITUDE data (X,Y order), and GEOMETRY stores geographic or projected coordinate system data (the exact coordinate system defined by the SRID assigned to the geometry). GEOGRAPHY has limits on extent (antipodal distance) that make GEOMETRY desirable even when the data is geographic (e.g., WGS 1984)
As Asrujit indicated, the keyword availability doesn't indicate data in both formats is present (in fact, it could be present even if the keywords weren't), keywords are a metadata property of tables created by ArcGIS (but because Query Layers can exist that aren't registered, the table properties of GEOGRAPHY data might not indicate a GEOGRAPHY keyword), and finally, a GEOMETRY keyword isn't required to create a GEOMETRY-based table (it could be any other keyword, or it could have GEOGRAPHY which was altered to create GEOMETRY tables). Basically, you can't rely upon the keyword if you haven't been the one in charge of all table creations.
- V