I have done that for cylindrical grinding jobs in the past, but generally, there are few jobs that CANNOT BE DOGGED, FINISHED ON ONE END AND TURNED AROUND AND COMPLETED WHER THE DOG WAS DRIVING.Winging it here... I suppose a different kind of dog might be used to turn the work using a radially mounted but axially oriented hole or two in the end of the shaft, using pins to drive the work without defacing the OD of the shaft. There would be an axial hole or two in the work. A 60 degree center would also be used as part of the tool to center the work, small diameter and used as shallow as appropriate. The tool would have to be shop made, but with mild steel it would be easy to machine, and there would be no slippage of the center of the work. THe pins could be standard hardened dowel pins. If the drive end of the shaft is big enough in diameter so the pins have some leverage, it would probably work just fine. The tool would not be difficult to make.
This is from my Audel’s “Progressive Machinist” c.1913 — it suggests the square centre is to be used more for correcting an inaccurately placed centre hole on the work rather than driving the work. At least if I’m reading correctly...
View attachment 308337
-frank
Unless the pin is perfectly located, it would tend to draw the part off center, I think better to have some clearance, that is what I did for grinding work.Using a drill bit on a mill with a centered rotary table would be my first idea. To make it even better, drill the hole(s) a little undersize and then ream them for a nice and snug pin to hole fit.