If $A$ and $B$ are square matrices, then it is known that $AB$ and $BA$ have the same eigenvalues.
Can the same thing be said about the spectral norms of $AB$ and $BA$? They have been the same in the few examples I've looked at.
If not true in general, I would be specifically interested in the case where $A$ and $B$ are orthogonal projections.
This is true, at least for normal matrices ($CC^* = C^*C$). In this case, you have that the singular values are $\sigma_i = |\lambda_i|$, so if the eigenvalues are the same, then the singular values are the same.
A counterexample for non-normal matrices:
$$ A = \begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}, B = \begin{pmatrix}1 & 1 \\ 1 & 0 \end{pmatrix}. $$
Then
$$ AB = \begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}, (AB)^*AB = \begin{pmatrix}1 & 0 \\ 0 & 0 \end{pmatrix}; \quad BA = \begin{pmatrix}0 & 1 \\ 0 & 1 \end{pmatrix}, (BA)^*BA = \begin{pmatrix}0 & 0 \\ 0 & 2 \end{pmatrix} $$
and the relevant singular values are, respectively, $1$ and $\sqrt 2$.