Cutting a Möbius strip down the middle Why does the result of cutting a Möbius strip down the middle lengthwise have two full twists in it? I can account for one full twist--the identification of the top left corner with the bottom right is a half twist; similarly, the top right corner and bottom left identification contributes another half twist. But where does the second full twist come from? 
Explanations with examples or analogies drawn from real life much appreciated.
edit: I'm pasting J.M.'s Mathematica code here (see his answer), modified for version 5.2.
twist[{f_, g_}, a_, b_, u_] := {Cos[u] (a + f Cos[b u] - g Sin[b u]), 
      Sin[u] (a + f Cos[b u] - g Sin[b u]), g Cos[b u] + f Sin[b u]};

With[{a = 3, b = 1/2, f = 1/2},
Block[{$DisplayFunction = Identity}, 
g1 = ParametricPlot3D[Evaluate[Append[twist[{f - v, 0}, a, b, u],
  {EdgeForm[], FaceForm[SurfaceColor[Red], SurfaceColor[Blue]]}]],
  {u, 0, 2 Pi}, {v, 0, 2 f}, Axes -> None, Boxed -> False];
g2 = ParametricPlot3D[Evaluate[Append[twist[{f - v, 0}, a, b, u],
  EdgeForm[]]], {u, 0, 4 Pi}, {v, 0, 2 f/3},
  Axes -> None, Boxed -> False];
g3 = ParametricPlot3D[Evaluate[Append[twist[{f - v, 0}, a, b, u],
  {EdgeForm[], FaceForm[SurfaceColor[Red], SurfaceColor[Blue]]}]],
  {u, 0, 2 Pi}, {v, 2 f/3, 4 f/3}, Axes -> None, Boxed -> False,
 PlotPoints -> 105]];
GraphicsArray[{{g1, Show[g2, g3]}}]];


Observe that the boundary of a Möbius strip is a circle. When you cut, you create more boundary; this is in fact a second circle.  
During this process, the Möbius strip loses its non-orientability. Make two Möbius strips with paper and some tape.  Cut one and leave the other uncut. Now take each and draw a line down the middle. The line will come back and meet itself on the Möbius strip; on the cut Möbius strip, it won't.
