What can be said about the quotient space associated with Peano's curve?

Question

Let $f:[0,1]\to[0,1]^2$ be a continuous surjection, e.g., the Peano's curve. What can be said about the quotient space $[0,1]/_\sim$, where $x\sim y$ iff $f(x)=f(y)$? I only know that the quotient space must be $T_1$, since $\sim$ is closed as a subset of $[0,1]^2$. Is it possible that the quotient space is homeomorphic to the square? Another question: is it possible that any two quotient spaces associated with continuous mappings of the interval onto the square are homeomorphic?