While trying to constructing a CW complex of Moebius band, I come up an construction which I am not sure about it. The construction is as follows: Attach two 1-cells to a 0-cell. Then attached a 2-cell in a way such that it starts at left half of upper circle,pass through 0-cell, arrived at right half of lower circle, then continued from left half of lower circle, to 0-cell, finally right half of upper circle. Is it a Moebius band, or something else?
Edit I should add details of the attaching map. By attaching two 1-cells to 0-cell, I mean the map $f_{1},f_{2}:S^{0}\rightarrow \{0\}$, $f_{1}(\pm1)=f_{2}(\pm1)=0$. Then we formed a "8" shape figure.
For the 2-cell, we attached $S^{1}$ by starting at the northpole of upper circle, then draw a "S" shape to southpole, then going back along the remaining part. So the concept is drawing a "8" from north pole.