TITLE: Is T2 modulo permutation by S2 a familiar surface?
QUESTION [5 upvotes]: If I have a 2d Torus $S_1 \times S_1$ but I identify $(x,y)\equiv(y,x)$, is that a familiar surface?
context: I am an engineering graduate student with a vanilla background in undergraduate topology pursuing a research question. I tried the usual gluing of edges but had difficulty embedding in 3 dimensions (which is my best chance at recognizing it). I felt the answer should be widely known on the internet but couldn't find it.

REPLY [6 votes]: Take the square $I \times I$ and form a torus, we have to take the quotient $(0,a) \sim (1,a)$, and $(x,0) \sim (x,1)$. Now, to make our identification $(x,y) \sim (y,x)$ we see that we can "fold the square" over diagonally (along the the line from $(0,0)$ to $(1,1)$.
But the order in which we take our quotients does not matter.
So, first make the "diagonal fold." Following the usual identication for the torus, we see that there are some orientation problems. To fix this, we can take another diagonal cut from $(1,0)$ to $(.5,.5)$. Remove both mini triangles and glue $(x,0)$ to $(y,1)$ together. Reiattaching the cut we made earlier, we see that the orientation has reversed,so this is just the mobius band.
This is difficult without a picture. Thankfully, 3blue1brown has an excellent video on just this around time 12:00 should give a very nice geometric intuition for why the answer is a mobius band (with beautiful animations)