2nd Degree Crossovers
Its not weave its not butterfly ... its more of a concept, so ill post it in everything else - Matt, please put it wherever you think is appropriate.
The Theory:
0 degree crossover: Spinning Poi at your sides.
1st degree crossover: Right Poi left plane, Left Poi right plane. Arms are crossed, obviously. BHB can get quite comfortable after some practice.
2nd degree crossover: Right Poi right plane, Left Poi left plane. Arms twisted like in a 5beat weave.
Most of the time one is passing through the 2nd degree crossover via 5beat weave, offset etc. but try to keep it spinning there without the Poi changing planes and gain control over it. It helps with complex turning and more.
Try splittime same direction first. Then sametime. The planes are not 100% parallel but its a matter of practice.
Butterfly in a 2nd degree crossover is one of those ¨forgot how it feels to hit yourself?¨ moves 
Windmill has a mix between 1st and 2nd degree and is therefore easier.
Greetings from Vancouver
andy
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2nd Degree Crossovers
I've been playing with something sort of similar to these, 5bt offset weave carries in the wall plane which are quite nice and odd-looking
Some more ideas here to play with though!
2nd Degree Crossovers
ive been experimenting with these aswell... this stuff makes me dizzy though. 8O
2nd Degree Crossovers
@spiralx: 5bt offset, carries etc are derivatives of 2nd degree crossovers. Try to play with the "pure" crossover, without plane changes and offsets ... should become easier and cleaner
2nd Degree Crossovers
andy, do you check the I just learned a new move thread on HoP.. I posted months ago.. could have really used some discussion then
I made some interesting weave and butterfly weave patterns from this.. I used it as a tool to straighten my planes.. My favorite was taking opposite spinning poi, and cross, 2nd cross, and back to 1, back to 0, to -1 to -2 back to -1 back to 0... you get this neat atom looking thing, helps with planes, works half circle changes, and gets you into some interesting positions that you wont get normal spinning..
then I found these are just as much fun if you spread them out.. instead of keeping it as this tight center fixed motions (because I did mine with my wrists together) you can use the motion kinda bouncing from side to side and make weaving patterns out of it..
fun stuff.. the horizontal versions are killer too.. that's what got me onto the 4-5bt antispin corks... hell I think I briefly mentioned these here too..
2nd Degree Crossovers
oooh, now someone will finally understand my spiders.
i've been working my way up to these as part of my inversions/insides/atomics discussion. but because of the "beat" ethos definitions i've yet to get there. thanks for saving me the trouble and now i have a set of words to sidestep the beat ethos! plus you've done a much better job at explaining it than i ever could
1 degree crossover= 3 beat spider
2 degree crossover= 5 beat spider
so now you may know what i meant at hop when i said there's ways to go from 3-5 beat atomics, i really mean going from 3 beat atomics/spiders (1 degree crossover) to 5 beat atomics/spiders(2nd degree)