Holding the sword
For combat use after the blade is drawn the left hand would hold the blade as far down as possible with the little finger curling around the butt cap (Kashira see above picture for sword part names) this is your power hand.
The right hand grips just below metal sleeve (fuchi). The right provides the control. But if you where to feel the cold steel of the fuchi during battle you would know that your hand slipped to far up the handle. If this occurred in battle you could break your finger from the force of your opponents blows running thought the tsuba (guard).
Correctly gripping the sword is as important to swordsmanship as forming a correct fist is for punching in boxing or karate. Gripping the sword is so important that, if not learned correctly, all other aspects of the sword will never be performed correctly.
When practising alone many students misunderstanding the actual purpose of the cutting action in preference to the result they feel they need to achieve in power and speed. This ultimately causes the grip to be practiced incorrectly, a mistake that is very hard to correct later.
Correct grip allows the distribution of impact up through the arms for either cutting or blocking (and not through the thumbs). Creates correct alignment between the blade, arms and body for correct cutting angles. Ensures that you can stop the blade from continuing beyond a safe cutting arc. Keeps the swords man a maximum distance from the enemy at the correct point for the cut.
The Katana is designed to cut sometimes this fact is lost in sweat and heavy breathing. As practitioners get tired, their hands can rotate until the palms are beneath the handle holding up the sword rather than being behind the blade to perform the cut. This bad habit has an additional negative effect in that it causes students to bend their arms to further support the thumbs and hands from the swords weight throughout the arc of the cut. They then lose the advantages gained by the correct grip.
Japanese swordsmanship is a whole life long study in its own right and beyond the scope of this work. A good book on swordsmanship is the “Deity and the sword” Katori Shinton Ryu by Risuke Otake