I have been wondering where, how and when you intend to have the transitions between different plane orientations.
Enter one door and reappear through another, in another orientation, or what? Have the avatar put one foot onto a wall and stepping right onto it? (implying need for a custom avatar animation). Twisting pathways?
Is the camera meant to follow the avatar around, or should we see ourselves walking upside-down?
One way is to use so called "subworlds". These are "simply" the locking of your avatar coordinates, to the coordinate system of another object, instead of directly to the world one. This is commonly used for vehicles; move the subworld (vehicle) and the avatar will move with it (without having to calculate transfer of momentum, or any other physics -- your position is simply relative to the subworld object's origo). These allow you to have "down" in whichever direction you wish (you'll warp around right away, upon switching to the subworld) and each have their own sets of colliders, so that it is e.g. possible to walk through a wall while in the reference space of one subworld, but not in another, if you leave that wall out of the one's collider mesh.
For a good example of subworlds, with source (.blend fle), you could look at the NadnerB's "AnimatedDoor" example age, here:
http://www.guildofwriters.com/wiki/Animations.
It has a platform, which you can step onto and click on, at which point it will travel up, turning over as it goes, leaving you literally standing on the ceiling. (Don't step off the platform at this point. Doing so will trigger your leaving the subworld and you'll return to the world coordsys.)
More about subworlds at:
http://www.guildofwriters.com/wiki/SubWorldsYou could easily trigger a subworld shift, using a "region" (an invisible volume, which can trigger any sequence of actions, upon entering or leaving it), placed right in front of a wall, optionally with an extra conditional that the avatar must be facing the wall, for the transition to occur, in case you don't want it to make the switch if the player backs or sidles up to it, or to avoid any possible risk of getting into infinite loops of the avatar stepping on-off-on-off-etc...
If you wish to be "extreme" you could rotate the entire world, so that down is always down... :)
EDIT1: Hmm, I see others had already said the same things I just did... :D
EDIT2: I notice that one of the "oneshot" avatar animations available to us is the "jump up on the spot" one. You could potentially use this for subworld transitions, if you move the avatar into an intermediate subworld and animates that subworld (rotating 90 degrees and, if necessary, translating it a bit) while the avatar is airborne -- it might actually look half decent; you jump up, rotate in the air and land on the wall...
(Done exacly this way, one might get mutiplayer issues, mind, with the player "taking another player along for the ride").
EDIT3: Incidently, I had a similar idea, back when the first age contest was held. Its rules stipulated that the age should be a maximum of so-and-so many blender units, cube, in size, which led me to the concept of an age literally consisting of a Blender grey view grid; three intersecting planes, exactly as large as the rules stated, giving you eight separated sections to visit, in different (dis)orientations. You'd need to travel between all these, with the help of various representations of Blender tools (like e.g. size yourself down to get through a small hole), until you got to the delicious cake at the centre, visible from all, but reachable only from one of the planes. :P
EDIT500: ...one more revision... *moves misplaced comment* ..ok... think I can go to bed now.. :P