I had a cylindrical grinder job in trade school that had to be turned around because of length and dogging, and got a parting line where the two grinds met in the middle..could not even measure/find the error, but it looked like crap. After I learned the ropes, I figured the headstock center was off.
I had a respected old-timer who told me "Whatever the error in the center point figure you will get double that on the part because it takes off one side and leaves on at the other side."
With thinking that I have solved grinder machine problems and have had great success in high-precision grinding.
I look for less than half center error comparing the specs on the part
To get a tenth on a part, I want less than 50 millionths on the center...
A 2 tenths error live center I regrind, or throw it in the dumpster.
Some China centers can be ground dead true because of poor bearings, bearing not set with ascetics lined up, or the diameters running with a wobble.
We had an expensive production grinder at the big shop that they bought a new spindle for a ton of money and it still failed..When I found that out I have them bring me the center..and I ground to near zero and the machine worked just fine.. likey there was nothing wrong with the old spindle.
Note: it is just about impossible to take apart a live center to regrind the point, that is what is wrong with China centers ..the process is wrong,
A live center spindle has to be ground between centers to be right, so all diameters run near zero..then you snub off the point end center and grind the point running in its own bearings.
*I have to laugh at YouTubers showing how they take apart a live center and regrind it.