Let $V,W \subset \ell_p^n(\mathbb{R})$ be subspaces of dimension $2$ or greater, and let $p$ be an even integer not equal to $2$. Is it possible for there to exist an orthogonal matrix $Q$ that maps $V$ to $W$ and satisfies $\|v\|_p = \|Qv\|_p$ for all $v \in V$?
This is trivial if $Q$ is a generalized permutation matrix, so I would like to know if this is possible for an orthogonal $Q$ that is not a generalized permutation.
Some related facts: if $p$ is not an even integer, then no such $Q$ exists. Also, if $Q$ is not constrained to be orthogonal, then non-orthogonal matrices satisfying these conditions exist for some choices of $V$ and $W$ (for example, if they are spanned by vectors with mutually disjoint support).