It is indeed possible; we claim rotating once around the point $\left(\frac{1}{2},\frac{1}{2}\right)$ works. Note that a rotation of the point $(a,b)$ 90 degrees counterclockwise around $(m,n)$ sends it to $(m+n-b,a+n-m)$. Thus, $(0,0)$ is sent to $(m+n,n-m)$, $(1,0)$ is sent to $(m+n,1+n-m)$, and $(0,1)$ is sent to $(m+n-1,n-m)$. Then, when $m=n=\frac{1}{2}$, $(0,0)$ is sent to $(1,0)$, $(1,0)$ is sent to $(1,1)$, and $(0,1)$ is sent to $(0,0)$ as desired. Thus, rotating around the point $\left(\frac{1}{2},\frac{1}{2}\right)$ works.