Tic-tac-toe on Mobius strip has something counter-intuitive. Please help!

I am not a mathematician.
I am trying to define tic-tac-toe on Mobius strip. (http://
I came across a problem while defining the diagonals. I need help.

Here is how I defined a Mobius strip board: (hopefully the picture
looks good)
| |
+>(0,2)|(1,2)|(2,2) >---|+
(0,1)|(1,1)|(2,1) | |
+>(0,0)|(1,0)|(2,0) >--+ |
| |

.... and (0,1) and (2,1) "roll around" to each other.

Horizontal "three in a line" conditions would mean any three
consecutive coordinates from below:
(0,0)-(1,0)-(2,0)-(0,2) notice it twisted - (1,2)-(2,2)-(0,0) twist
and back - and so on
and (0,1)-(1,1)-(2,1)

If I define "right and up" diagonal, one of them could be: (1,0)-(2,1)-
But if I define "up and right" diagonal, one of them could be: (1,0)-

That means the operations "up" and "right" are not symmetric to each
other(?) anymore. I coming across two diagonals originating with two
common points.

Which one of the above definition for diagonal is correct?