Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the two transformations $L$ and $R$, the rotations by $90^\circ$ counterclockwise and clockwise, respectively, about the origin. How many sequences of $6$ transformations chosen from $\{L, R\}$ will send all of the labeled vertices back to their original positions?

1. If $k$ of the six transformations are $L$, the net rotation is $90^\circ k-90^\circ(6-k)=90^\circ(2k-6)$ counterclockwise.
2. The vertices return to their original positions exactly when this is a multiple of $360^\circ$, i.e. $2k-6$ is a multiple of $4$, i.e. $k$ is odd: $k=1,3,5$.
3. The number of sequences is $\binom61+\binom63+\binom65=6+20+6=\boxed{32}$.