I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this.

Taking $n \le m$, one can show that $A \in \mathbb{R}^{m \times n}$ is an isometry from $\ell_p^n(\mathbb{R}) \to \ell_p^m(\mathbb{R})$ if and only if the columns of $A$ have unit $\ell_p$-norm and only one nonzero entry per column. This characterises the subspaces isometric to the subspace $\ell_p^n(\mathbb{R}) \subset \ell_p^m(\mathbb{R})$.

Also, for $p$ not an even integer, I believe all the isometries between subspaces of $L_p([0,1])$ have been classified, and this can likely be extended to $\ell_p^n(\mathbb{R})$.

Does anyone know any references in this direction for even $p$?