Examine each pivot in turn with the help of an eyeglass for signs of wear. Pivots should have highly polished parallel sides. If no lathe is available, one method of re-furbishing is to lightly fit a pinvice to the other end and rest the pivot to be polished on a block of wood with a groove cut in it. The pinvice is rotated while a pivot file is rubbed over the pivot. Once an even finish has been obtained, a burnisher is applied in the same manner to bring it up to a mirror finish. A new burnisher is often difficult at first and improves with a cross grain applied using medium grain emery paper. Place the emery on a flat surface and pass the burnisher over it a couple of times.
Repairing broken pivots normally needs the use of a lathe but pivot caps can be obtained.
Repairing Your Own Clocks by Mervyn Passmore