Can these phenomena occur within Non-Euclidean geometries? I've enrolled in an undergraduate seminar on the subject of non-euclidean geometry. I wanted to ground myself a little before-hand, because popular media has lead me to believe that non-euclidean geometries are closely related to portals, witch-craft, and Cthulu.
In particular, the term is frequently invoked to describe phenomena that occur in the game Antichamber. Wikipedia has this to say about Antichamber:

Many of the puzzles are based on phenomena that occur within the
  Non-Euclidean geometry created by the game engine, such as passages
  that lead the player to different locations depending on which way
  they face, and structures that seem otherwise impossible within normal
  three-dimensional space.
Puzzle elements in various chambers involve maneuvering themselves
  around the non-Euclidean spaces, where level elements can change after
  passing certain points, or even based on which direction the player is
  facing when traversing the level.

see: http://en.wikipedia.org/wiki/Antichamber
How much of what we see in the media is an exaggeration of the implications of non-euclidean geometry? Are there only two kinds of non-euclidean geometries (elliptic and hyperbolic), or can an arbitrary space with extra rules governing movement be considered a non-euclidean geometry. It seems to me that a person's facing while moving in a space shouldn't have any bearing on there position, regardless of whether parallel lines meet at infinity... is there some tricky surface on which this is possible?
 A: "Non-Euclidean" gets tossed around a lot in popular culture, I would guess due in large part to the influence of H. P. Lovecraft. Its meaning in popular culture is extremely broad and I wouldn't spend too much time trying to match it up to the mathematics.
In mathematics it has, I suppose, a few possible meanings. There's a 


*

*very narrow sense in which it refers to either elliptic geometry / spherical geometry or hyperbolic geometry, the two traditional examples of models of the first four of Euclid's axioms which don't satisfy the parallel postulate;

*broader sense in which it might refer to a model geometry; although there are only three model geometries in $2$ dimensions, namely Euclidean, spherical / elliptic, and hyperbolic, in $3$ dimensions there are eight;

*very broad sense in which it might refer to any Riemannian manifold.


The kind of stuff that happens in Antichamber, though, has more to do with topology than geometry. Roughly speaking, geometry usually means something like angles and lengths are involved, whereas something like portals or rooms related in unexpected ways have more to do with how the different parts of the space you're moving around in are put together, which is more of a topological notion. 
In particular, perhaps the most important ways in which non-Euclidean geometries differ from Euclidean geometry has to do with nontrivial curvature, which is a geometric rather than a topological notion, and nothing that I remember happening in Antichamber has anything to do with curvature. 
(It's worth noting that "non-Euclidean" is a term of primarily historical importance, and it's not the sort of term mathematicians would use while talking to other mathematicians, much like "fractal." Nowadays calling something non-Euclidean is like calling something a non-banana.) 
