Identifying a $\Delta$ complex
Solution 1:
You get a cone if you identify the edges $[v_0, v_1]$ and $[v_1, v_2]$ reversing the ordering of the vertices (because you identify $v_0$ with $v_2$ and $v_1$ with $v_1$ when you do this).
To preserve the orientation, you need to make a "twist" to identify $v_0$ with $v_1$ and $v_1$ with $v_2$. This gives a Möbius band.
You can see more details (and pictures) here.