orientability
Orientability is the property of a surface having a consistent notion of 'this side' versus 'that side', or equivalently a consistent sense of 'clockwise'. A sheet of paper is orientable: it has a front and a back, and if you draw a little clockwise arrow on it and slide that arrow around anywhere on the surface and bring it home, it is still clockwise. The Mobius band is the famous failure: take a strip, give it a half-twist, and glue the ends, and now an ant can crawl from any point to the 'other side' without ever crossing an edge — there is only one side, and a clockwise arrow comes back reversed.
Precisely, a surface is orientable if you can choose, at every point, a consistent sense of rotation (or a consistent choice of 'outward normal') that varies continuously and matches up everywhere as you move around any closed loop. It is non-orientable if somewhere on it there is a loop along which sliding your little circling arrow flips its handedness when it returns — an 'orientation-reversing loop'. The Mobius band's centre circle is exactly such a loop. A surface is orientable precisely when it contains no Mobius band hidden inside it; non-orientable surfaces are exactly the ones that do.
Orientability is one of the two numbers that classify compact surfaces (the other being genus): the orientable ones are the sphere and the n-holed tori, the non-orientable ones are the projective plane, the Klein bottle, and their relatives. The honest caveats: non-orientable surfaces are perfectly real mathematical objects, not paradoxes — the Mobius band sits happily in our three-dimensional world. But the closed non-orientable surfaces, like the Klein bottle, cannot be built in three-dimensional space without passing through themselves; that self-crossing is a limitation of three dimensions, not a flaw in the surface, which lives cleanly in four dimensions.
Make a Mobius band from a paper strip with one half-twist, then draw a line down its middle with a pen, never lifting the pen, until you return to the start. You will find you have drawn on 'both sides' in one continuous stroke and the line is twice as long as the strip — proof there is really only one side. Cut along that centre line and, surprisingly, the band does not fall into two pieces but stays in one longer loop: orientability failing has tangible consequences.
The Mobius band: one half-twist destroys two-sidedness, and a midline cut leaves it in one piece.
A closed non-orientable surface like the Klein bottle cannot embed in three-dimensional space without self-intersecting; the self-crossing is an artefact of cramming it into 3D, not a defect — it embeds without crossings in four dimensions.