The Sageo of Katana
The sageo is a cord attached to the scabbard (saya) of the sword by the (kurikata). Its characteristics can be variable: shape (round or flat), length (short or long) and material (cotton, silk, leather).
During the Kamakura period (1185-1333) the samurai was still essentially a horseman and his sword, the tachi was perfectly suited to this situation. It was carried sharp down and hung from the belt by the sageo.
Later, during the Edo period (1600-1868) the historical and social evolution of the charge of the samurai made him a pedestrian and modified his weaponry. The tachi was replaced by the daisho (couple katana / wakizashi) and this diptych was carried sharp upwards and sheaths pushed in and held by the belt (obi). Despite the loss of its original vocation, the sageo was preserved but its role underwent a change.
The Sageo of yesteryear
On the battlefields the samurai used the sageo as a strap to attach the scabbard to his belt and prevent it from being lost in combat.
When the situation permitted, the samurai detached the sageo from the kurikata and used it to tie up the wide sleeves of his kimono and thus not be hampered when handling his sword.
The sageo was also detached from the scabbard to be used in binding techniques (hojo-jutsu) of captured enemies.
The sageo provided a string availability for all situations where this kind of material was necessary (repairing sandals for example). As a Japanese proverb (kotowaza) indicates: "being prepared is the guarantee of freedom from worries".
Note that in the Edo period, the samurai who became pedestrians were attached to Houses where each movement was formalized. In this setting where the saber (katana) frequently had to be put down, the sageo was not tied to the belt but simply placed behind the scabbard or even wrapped around it.
In the context of the Muso Shinden Ryu school, the sageo is slipped into the obi on the right part of the hara, taking care to let a sufficiently large part take not to hinder the mobility of the saya.
Each ryu has its approach and sometimes differences appear within the same school. The sageo can be tied on the left side, slipped behind the scabbard, wrapped around the scabbard, etc ... Some styles even do without the sageo!