Kwijibo's name originated, most likely, from the first episode of the Simpsons. How the spelling changed is anyone's guess. The trick looks somewhat simple, but is incredibly hard to do. It is reminiscent of "off-string" yo-yoing, and that is what gives its instantly recognizable effect. While insanely hard to learn (for most, at least), no yoyoers repertoire should be without it. Patience will pay off here, as this is a fundamental new-school trick.

This trick is illustrated as if you were looking into a mirror. 

Step 1

Start with a Trapeze

 

Step 2

Steps 2 through 4 should be done in one smooth, almost instant, motion. 

Pop the yo-yo off the Trapeze to your right, as indicated by the orange arrow. Move your right hand in the direction of the blue arrow.

Step 3

You should be in this position. Your right hand should continue to move to your left and your left hand should start to move to the right. You are basically crossing your hands. Pay particular attention to the red string segment between your hands. The yo-yo will need to land on there as a result of you moving your hands in the direction of the arrow.

 

Step 4

You should be in some position similar to this. You want your left hand, however, to be where the blue X is so if it's not there, move it there by following the direction of the blue arrow.

 

Step 5

Before freaking out, ignore all the arrows and focus on the string formation. Is it similar? If so, then you did the first "catch" correctly.
Now, let's deal with the arrows. All this basically means is to flip everything 360-degrees clockwise. You should then end up in a 1 and a half mount.

 

Step 6

This is a one-and-a-half mount, although it doesn't look like it too much (if you moved your left hand up in this picture it would look normal. Anyway, it is time to pop the yo-yo up one more time. The next series of steps should be done as one fluid motion. It should look almost instant.

 

Step 7

After popping the yo-yo up, it will be airborne as shown in the picture to the left. You want to move your left hand in the direction of the arrows. However, you must, when you follow the path of the arrows, hit the red string with your left hand pointer finger. This is hard to do.

 

Step 8

This picture is  in "mid-step" of the arrow path from Step 7. In other words, when you hit the red string as said in Step 7, your formation (for a split second) will look like this. Continue moving your left hand in the direction of the blue arrow to the blue X.

 

Step 9

The yo-yo, since it was hooked by your left finger will also have moved to the left  and now it is about ready to fall onto the red string segment as indicated by the orange arrow.

 

Step 10

You should now be in the double or nothing position. Now, everything up and land it in a trapeze, then end with a fly away dismount and you're finished. 

Note, when performing this trick should look like: Trapeze, toss up, catch in some funky hold, flip everything over to a 1.5 mount, toss up, catch in a double or nothing.