Katanas are not holstered upside down, but they are typically worn with the cutting edge facing upward when sheathed. The way katanas are worn and sheathed is a result of practicality, accessibility, and traditional customs. Here's why katanas are worn with the cutting edge facing upward when sheathed:

1. Drawing Efficiency: Wearing the katana with the cutting edge facing upward allows for a more efficient and fluid drawing technique. The samurai would grip the hilt with their dominant hand and use their thumb to push against the tsuba (handguard) to loosen the sword in the scabbard. This grip and motion facilitated a swift and smooth draw, enabling quick deployment of the sword during combat.
2. Protection of the Blade: Placing the cutting edge facing upward inside the scabbard helps protect the blade from accidental damage or dulling. When the katana is sheathed, the back of the blade rests against the inside of the scabbard, preventing contact between the sharp edge and the scabbard. This orientation helps maintain the sharpness and integrity of the cutting edge.
3. Traditional Custom: The wearing of katanas with the cutting edge facing upward has been a long-standing tradition and cultural practice in Japan. It was an integral part of the samurai's code of conduct, known as bushido. Adhering to this tradition demonstrated respect for the sword as a symbol of honor, and it reflected the samurai's readiness for combat.

It's important to note that the specific method of wearing the katana, including the positioning and angle in relation to the belt or sash (obi), may vary slightly depending on the specific style or school of swordsmanship. The way the katana is worn and sheathed may also have symbolic and cultural significance within the context of different martial arts traditions.