Roger, the flats are for driving dogs, and you might find size and perhaps other info stamped or etched on the flat surface. It is possible, with care, to grind oppposite surfaces by turning the mandrel and work end-for-end, to avoid any changes to the work's position with regard to its centre line. Given the very slight taper, for precision work, the workpiece could cant very slightly, and opposite faces would not be parallel. Having said that, when I was "bossing and facing" the ends and registers of cutters, I pressed the mandrel out, turned the work around, and pressed the mandrel in again.
I have a vague memory of one end sometimes having a ring (or am I thinking plug gauges), but certainly one end was often shorter than the other.
The [driving plates/dogs, catch plates] I used were lighter than the lathe types, and with arms off both sides, similar to the third picture here:
https://katalog.mav.cz/categories.php?rozbal=3443
This reduces any imbalance in the work, which might otherwise show up in the roundness/concentricity of the work.
Here is an example with flats on both ends:
http://www.ebay.com/itm/TTC-Lathe-And-OD-Grinder-Mandrel-Diameter-1-3-4-Overall-Length-10-/231948753418?hash=item3601391a0a:g:A3oAAOSwcvdXPHvJ
My Machinery Handbook (17th Ed., 1964) mentions mandrels in the index, but calls them arbo[u]rs in the text, and recommends 0.006" per foot for turning. Each end is shown the same length, and it doen't mention their use for grinding.
My Workshop Technology Pt (Chapman, 1968) shows the steps in grinding a headed bush, by grinding the bore, then, on a mandril, grinding the ODs and faces on one side, reversing mandrel and work, then facing the other end.
Bill