I need this pipe center maybe twice a year and have hitherto managed to work around some nasty gouges and runout that came with it. But I finally cleaned it up by the simple expedient of dialing the compound to match the average angle (it was badly out of true) and skimming off enough to clear the surface.
The trick was realizing that the teeny little shoulder between the taper and the cone was concentric with both. I grabbed it tight (yeah, in a three-jaw chuck), took sissy cuts, and hit the end result with a file to smooth things out.
The remaining gouges are just fine by me.
It had been center-drilled in the small end, but the opening had taken a real beating at some point. Neither the live nor the dead center sat correctly and I couldn’t figure out how to hold the thing to re-drill the end.