I believe I have a general understanding of the mechanics of how this stage flips now. I did several trials in training mode to conclude what I am posting. I am sure of one thing though; Frigate Orpheon's flipping is 100% non-random (when it flips is random of course, but the RESULTS of the flip are completely predictable).
*This is just a THEORY based off the evidence I have witnessed. Alternative and more scientific testing could be interesting and perhaps supplant my theory.*
The stage actually spins; think of the other half as always existing above you, and when it flips, it's like being in a centrifuge. You will always end up at the same coordinate you started unless an object (either the form you start on or the form you land on) gets in your way. The first form always spins to the left (clockwise) while the second form always spins to the right (counterclockwise). Your momentum when it starts spinning is conserved so moving before it spins can result in you being flung in quite the odd way, but basic physics could always predict where you end up (given that brawl uses havok, I bet it's accurate).
As per useful hints to not ending up under the stage, here's what I can offer:
Being in extreme positions results in a net non-change. If you are very high up, you shouldn't come in contact with the stage at all and should translate to the same position. Being very low works the same way but is obviously unsafe for most of the cast.
The left side of the first form and the right side of the second form should be the safer sides. If you are on these sides, the form you start on won't "drag" you as much when it flips.
The only way you are going to get caught under the stage is if you enter the second half below it. Higher elevations are safer for sure; the dojo's advice is rock solid in this regard.
Not being moving on the left platform on the first form ends with you standing on solid ground 100% of the time.
About the report of being star KO's when it flipped, I have only one explanation. If you found just the wrong position, you might get "snagged" on the form of the stage you are flipping away from. If you did get stuck as such, surely you would be killed off the top. It's incredibly unfortunate and incredibly unlikely (I have not managed to reproduce it), and the only way it will happen is if you are on the wrong half of the stage when it starts flipping.
*This is just a THEORY based off the evidence I have witnessed. Alternative and more scientific testing could be interesting and perhaps supplant my theory.*
The stage actually spins; think of the other half as always existing above you, and when it flips, it's like being in a centrifuge. You will always end up at the same coordinate you started unless an object (either the form you start on or the form you land on) gets in your way. The first form always spins to the left (clockwise) while the second form always spins to the right (counterclockwise). Your momentum when it starts spinning is conserved so moving before it spins can result in you being flung in quite the odd way, but basic physics could always predict where you end up (given that brawl uses havok, I bet it's accurate).
As per useful hints to not ending up under the stage, here's what I can offer:
Being in extreme positions results in a net non-change. If you are very high up, you shouldn't come in contact with the stage at all and should translate to the same position. Being very low works the same way but is obviously unsafe for most of the cast.
The left side of the first form and the right side of the second form should be the safer sides. If you are on these sides, the form you start on won't "drag" you as much when it flips.
The only way you are going to get caught under the stage is if you enter the second half below it. Higher elevations are safer for sure; the dojo's advice is rock solid in this regard.
Not being moving on the left platform on the first form ends with you standing on solid ground 100% of the time.
About the report of being star KO's when it flipped, I have only one explanation. If you found just the wrong position, you might get "snagged" on the form of the stage you are flipping away from. If you did get stuck as such, surely you would be killed off the top. It's incredibly unfortunate and incredibly unlikely (I have not managed to reproduce it), and the only way it will happen is if you are on the wrong half of the stage when it starts flipping.
Very interesting info. 
