The noticeable difference between the headstock centre and the tailstock centre is their length. The tailstock centre is longer than the headstock centre to allow easy ejection from the tailstock. Should the headstock centre be put in the tailstock by mistatake it would be difficult to eject.

Fig.1 displays the method used to check the accuracy of the headstock centre using a (DTI) Dial test indicator.
Fig.2 shows a soft centre being turned true with the compound slide set at 30 degrees to the machine's axis.
A tool post grinder is used in place of the cutting tool when truing a hardened tailstock centre.