Q:

Why is this not a well-defined $\Delta$-complex of the torus?

My lectures notes say that the second diagram isn't a well-defined $\Delta$-complex of the torus because the $2$-simplices aren't totally ordered.

I don't really understand what that means.
Let's call the upper $2$-simplex $U$, the lower one $L$, the vertical $1$-simplex $a$, the horizontal one $b$ and the diagonal one $c$. 
If we orient the top $2$-simplex clockwise and the bottom one counter-clockwise. Then the boundary of the top one is $a+b+c$ and the boundary of the second one is $-a-b-c$. 
So where's the problem? Why is the first diagram well-defined, and the second one not? 

A:

Let's ignore the side identification for a bit, and just take a look at the following picture, where the orange arrows represent the orientation of a line (a $1$-simplex):

On the left-hand side we see that we can label the vertices $v_i$ in such a way that the numbering agrees with the arrows.
That is, if we go along a line in the direction of an arrow, then we go from a smaller number to a bigger one, and vice versa.
This corresponds to the fact that we require that the restriction of $n$-simplices to their faces ($n-1$-simplices) must respect the ordering of the vertices ($0$-simplices).
Try labelling the diagram on the right in the same way -- see if it's possible.

A:

For a $\Delta$-complex the orientation of each simplex is given by a linear order on its vertices, the faces then have to be oriented accordingly. Hence, the $1$-faces of a simplex can never be oriented in a cyclic manner.

