Given a orthogonal matrix $Q \in \mathbb{R}^{n\times n}$, we know $Q^{\top}$ is also orthogonal. Let $Q$ represent a linear transformation from an Euclidean space to itself, then reading from the columns and the rows of the matrix $Q$, we obtain two sets of orthonormal basis.
Are there any relations between these two sets of orthonormal basis either from the algebraic or geometric perspectives?
I cannot think of anything concrete. Perhaps $Q^{2}$ tells us something?