Smashing two planes together?

Minh-Tam and I were just discussing algebraic topology, and now we're wondering the following: what do you get when you smash two planes together?

(By that, I mean $\mathbb{R}^2 \wedge \mathbb{R}^2$, where $\wedge$ denotes the smash product.)

Stay tuned! (I will figure out the answer.)