Two Coordinate Systems on the Mobius Strip

x[u_, v_]  := 
       {(2  -  v Sin[u/2]) Sin[u],  (2 - v Sin[u/2]) Cos[u], 
         v Cos[u/2]}

y[u_, v_]  := 
       {(2  -  v Sin[Pi/4 + u/2]) Cos[u], 
        - (2 - v Sin[Pi/4 + u/2]) Sin[u],  v Cos[Pi/4 + u/2]}

ParametricPlot3D[x[u, v], {u, 0, 1.9 Pi}, {v, -1, 1}]

ParametricPlot3D[x[u, v], {u, 0, 1.99  Pi}, {v, -1, 1}]

ParametricPlot3D[y[u, v], {u, 0, 1.9 Pi}, {v, -1, 1}]

ParametricPlot3D[y[u, v], {u, 0, 1.99 Pi}, {v, -1, 1}]