Scissor Blades

The Scissor Blades (片太刀鋏), known as the Rending Scissors (断ち斬りバサミ) when combined, are a giant pair of scissors developed by Isshin Matoi specifically for cutting Life Fibers, that were divided into two weapons shortly before his death. They are made from hardened Life Fibers, making them the only weapons (beside Satsuki Kiryūin's Bakuzan) that are capable of cutting any Goku Uniform or Kamui, except for Ragyō Kiryūin's Shinra-Kōketsu, which is also made of hardened Life Fibers.

It is unknown what happened to the Scissor Blades after Episode 24, as Ryūko is not shown bringing them back to Earth with her.

Red Scissor Blade
Ryūko Matoi owns the red half of the Scissor Blade. The crescent-shaped hilt can extend into a longer two handed sword, known as Decapitation Mode. Ryuko's signature finishing move is Fiber Lost (繊維喪失 Sen'i Sōshitsu), a long-range forward thrust that completely destroys a Goku Uniform, allowing her to absorb the Banshi.

Purple Scissor Blade
Nui Harime formerly owned the purple half of the Scissor Blade. Unlike Ryūko's half, it has a circular handle. It can change into a dual scythe mode with a scythe at either end. It was eventually recovered by Ryūko in Episode 22 during the fight against Nui, officially completing the former's weapon set.

One half of the Scissor Blade seems to have changed color at some point, as it is shown to be red during Isshin's murder, purple when in Nui's possession, and red once again when Ryūko retrieves it.

Rending Scissors
When the Scissor Blades are used together, they are able to sever Life Fibers as well as prevent their regeneration, as shown by Nui's loss of her eye and arms (done by Isshin/Sōichirō and Ryūko respectively). Even though the Rending Scissors need to be used together to activate this effect, they do not need to be combined together to use this effect meaning that it can have the same effect even when it is divided into two halves. Also, since Ryūko demonstrated the ability to change the red scissor blade into a normal scissor half, it is presumed that she can also change the both the Rending Scissors into a normal sized scissor.

In Episode 23, both the Scissor Blades go into "Decapitation Mode", so it is presumed that the Red Scissor Blades can use the "Dual Scythe Mode", too. In Episode 24, the Scissor Blades were once again bolted together and shown to go into an "Duel Decapitation Mode."

Trivia

 * The boss character Mighty No. 7, from the upcoming video game Mighty No. 9, wields two halves of a giant red scissor thar bear a striking resemblance to the one wielded by Ryūko, indicating a possible reference to Kill la Kill.