Most cabochons can be re-polished while still in the mounting.
Because there is always some risk in removing a gemstone from the mounting, it is safer to be able to re-polish the stone while it is still in the mounting, if possible. It also saves the cost of un-mounting and re-setting the gemstone.
The ability to re-polish can be limited by obstructions near the stone. There is also a limitation on how close to the mounting bezel we can re-polish. Getting too close could damage the bezel. The majority of the time we are able to improve the appearance of the stone greatly with very little risk to either the gemstone or the mounting.
See examples below of Cabochons we have Re-Polished while still in the mounting