Scissor Blades

The Scissor Blades (片太刀鋏) are special weapons developed specifically to cut Life Fibers. They were created by Isshin Matoi as one single pair of scissors, the Rending Scissors (断ち斬りバサミ), and were divided in the moment of his murder.

Red Scissor Blade
Ryūko Matoi owns the red half of the Scissor Blade. It is has a elongated triangle handdle and can expand into a longer two handed sword (known as Decapitation Mode) or collapse into a normal size scissor half.

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.

This half of the Scissor Blade seems to have changed color at some point, since it is shown to be red during Isshin's murder. It then turned red once again when recovered by Ryūko.

It was eventually recovered by Ryūko in Episode 22 in order to fight against Nui, officially completing the former's weapon set.

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 blade can use the "Dual Scythe Mode" too.

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 the Kill la Kill series.
 * It is unknown why the blades didn't change color when both were in Nui's possession, and did change when Ryūko retrieved it without having to modify the color.