What is realted to invincibility?
I think that the reason to why it only happens with invincibility is that invincibility turns spikes into walls (according to Terry) and that if there's only one tile of distance between a wall and a scrren edge, a collision error happens (
slides through the wall upwards and flipped, though he/she flips back when the top of the wall is reached).
It's obvious it happens only in invincibility mode, because if it is off, you just die if you attempt to do this.
If spikes weren't solid like in The Tower, it wouldn't happen, unless there isn't space between the spike and the screen edge.
Anyway, Terry has recommended various times to players to not use invincibility, so why did he create it? To help beginner players?
Why not? It can be very useful for level creators, for frustrated people, for people who want to have a new game mode (speedrun with invincibility on) or people who love game glitches
Just like the unlock menu, it can be useful to avoid frustration if the save data becomes corrupted and you lose everything, then you can still unlock things you worked hard for to get.
For level creators there are 3 options:
- Check how hard the level is at the same time you check the scripts.
- Create a checkpoint next to the script and erase it later.
- Use slowdown mode (I believe it doesn't make glitches, if it does, then don't do this).
I recommend the 1st one. If you really don't want to do that, do the 2nd one. If you really don't want to do that, do the 3rd one.
For frustrated people there's slowdown mode, but I believe it's better to play fairly in "normal" mode and handle dying thousands of times.
Speedrun with invincibility? ... That might be a good reason for using it, but I'm not interested.
Why would anyone create a game mode only for people that like finding glitches?
Also, this is the first game If ind with an option like this since the beginning. I've seen games with a similar option, but only appears after dying a few times per stage, and it only appears in that stage. Perhaps that doesn't make much difference because players would only use it after dying lots of times.
The unlock menu would be helpful if you lose your save data or find a glitch like having 20 trinkets and not having unlocked the Final Level Time Trial. Otherwise, it would be cheating. I actually cheated accidentally while exploring that menu by unlocking No Death Mode. I actually tried it a few times, but only satrted trying it frequently after getting 4 S ranks.
The tower works normally, only Teleporter Divot, Seeing Red and Building Apport have no walls 
They probably turned into invisible walls. I think that those are available in Direct Mode. But, why did the spikes turn into those too?
I found a similar bug (that end up with the same thing) some months ago:
1: Same
2: Same
3: Play the final level until The Final Challenge.
4: At the rightmost pit with spikes, you'll go "behind" the upper part of The Final Challenge
5: Walk left ~2 Viridian lengths
6: Flip. Stay in the same area.
7: When you see Viridian above the spikes (you'll see his feet), flip to the ceiling again.
8: Walk right a pixel or two, and flip again. Continue walk right *all the time* outside The Final Challenge
9: You end up at the same place. Continue like Dav999 said.
I tried that several times, but I always end up in Outer Space.
You are *supposed* to end up there. Continue right and don't let go! Also make sure you fall DOWN. :p
How can you get there by falling down? Also, perhaps there's a way to get to Int.1. by faling down from there. You would stay in Int.1., because the rooms wrap vertically.
EDIT: Also, when using this glitch in the 3DS room, the topmost room (Whee Sports) is renamed "Copyright Infrigment". Maybe an AP-check that triggers by mistake, but there're no other things happening so.. :p
What's the relation between that and the glitch? What's an AP-check?
Also, perhaps this bug has been reported, as I remember seeing weird screenshots related to an invincibility glitch. I'll post a link later.
A new reply! I expected that, since I saw, in the "Who's Online" list, that Dav999 was quoting a post in this thread.
If you are in the room Tunnel of Terror, try to end up between the gravity lines of the Gravitron. I even messed up
's hitbox by doing that, and I can activate checkpoints, teleporters and terminals by just standing on the same height as the checkpoint/teleporter/terminal so if you enter a room from the left with a checkpoint at the right side of the room you activate it! Also, in
's room in the ship, you activate all terminals at once, and you can't leave that room anymore! Sometimes
refuses to go to the room on the left, and you can even stand on moving platforms if you're not, and be moving up and down in mid-air! Also, you can go through 1 pixel gaps in walls instead of 2 pixel gaps, because you're only one pixel wide instead of two! 
Those glitches have been seen before in the 3DS version and reported. That person even made videos of them!
Also, if you escape the Gravitron with slowdown mode, you can go through the hole in "Whee Sports" (or "Copyright Infringement"), ending up ina a room called "Outer space" that warps in all sides. I saw this in that person's videos. It was also seen before in v.1.2 and reported. Tha makes me think that

's hit box (I suppose that's the shape that if it touches spikes,

dies) is changed to make the Gravitron easier (I'm not the 1st one that thought about this). EDIT: I don't know why it activates all the terminals/Cps with the same y coordinate as

at the same time, though. END EDIT.
EDIT: Also, when using this glitch in the 3DS room, the topmost room (Whee Sports) is renamed "Copyright Infrigment". Maybe an AP-check that triggers by mistake, but there're no other things happening so.. :p
I heard that room was renamed because Nintendo didn't accept the game because of that room name. They thought it looked too much like 'Wii Sports'! 
Maybe they though that it was a reference to Wii Sports. It probably had to be changed even if it wasn't, though, because it was similar to "Wii Sports". However, they created an original room name!