Background
Consider the matrix $R \in \mathbb{R}^{12 \times 6}$ whose structure is given below as:
This represents a discrete gradient operator for a $2 \times 3$ grid equipped with reflexive boundary conditions. Then, the associated matrix $R^T R \in \mathbb{R}^{6 \times 6}$ (a discrete Laplacian) has structure given below as:
The matrix $R^T R$ can be written as the sum of BTTB + BTHB + BHTB + BHHB matrices, with "B = block(s)", "T = Toeplitz", "H = Hankel". Furthermore, since $R^T R$ is a symmetric matrix comprised of symmetric blocks, the matrix $R^T R$ can be diagonalized as \begin{align} R^T R = P_{\text{DCT}}^T D P_{\text{DCT}} \end{align} where $P_{\text{DCT}}$ is an orthogonal matrix representing the 2-dimensional Discrete Cosine Transform (DCT), and $D$ is a diagonal matrix.
Question
Now consider the matrix $R R^T \in \mathbb{R}^{12 \times 12}$ where $R$ is given as before. This matrix has a structure that is given below as:
Do matrices of this form have any special name, and is there a way to determine a diagonalization for $R R^T$ by making an observation about its structure (like we can for $R^T R$)?
Part of my trouble is that the off-diagonal blocks are neither symmetric nor toeplitz (due to the extra row of zeros at the bottom of each block). It is not clear to me that embedding the matrix within a larger one could help either. Experimentally, I do not seem to be able to diagonalize $R R^T$ with a DCT or a DFT. Perhaps there is a way to obtain a nice diagonalization by first permuting the entries?
Example on a larger grid
If it helps, here is what each of the matrices $R \in \mathbb{R}^{64 \times 32}$, $R^T R$, and $R R^T$ look like for a larger $4 \times 8$ grid:
