One of the most important things of the Japanese sword is the polish. A bad polish can ruin a sword whereas a good polish can raise the value of the sword.
To fully polish a sword you need to work up through at least 6-8 stones. First comes the basic shaping, the the refining of the basic shape and lastly the 'make-up' of the steel. Each next stones becomes finer and will remove the scratches of the previous stone. Also each stones has unique features that prepares the steel for the next stone constantly stacking unique aesthetic details on to op the next.
We speak of traditional polish when at least the last few steps in the polish are done with all-natural stones (nagura-do, uchigumori-do, hazuya, jizuya).
Also it's important that no acids are used to raise the contrast of the habuchi (temper-line) or hada (folding lines) or whiten the hamon.
The general rule is that synthetic polishing materials will ALWAYS make a surface shiny. Japanese (natural) polishing stones have the unique ability to polish a surface matte.
Whilst low-cost swords and cheaper re-production swords often have a shiny surface, a traditionally polished sword will always have a matte surface of the ha (cutting surface).
Note that the mune (back) as well as the shinogi-ji or bo-hi (area from the side ridge line towards the back) are supposed to mirror shiny. However these parts burnished into a mirror appearance instead of polished.
A general rule of thumb is that a proper polish will cost you about 30-50% of the value of the sword.