Given $3$ matrices:
$$\begin{aligned} A_1 &= \frac12 \left( X'X'^T + CX'X'^T + X'X'^TC + CX'X'^TC \right)\\ A_2 &= \frac12 \left( X'X'^T - CX'X'^T - X'X'^TC + CX'X'^TC \right)\\ A' &= X'X'^T \end{aligned}$$
where $X'$ is arbitrary and
$$C=\begin{pmatrix}J_{m/2} & 0 \\ 0 & -J_{m/2}\end{pmatrix}$$
(see https://en.wikipedia.org/wiki/Exchange_matrix for $J_n$).
Is there anything to be said about the eigenvalues and eigenvectors? It seems to hold that eig($A_1$)+eig($A_2$)=eig($A'$) and additionally $V'$eig$(A') = V_1$eig$(A_1) + V_2$eig$(A_2)$ in some examples I've tested. However I can't seem to prove the fact.