Let $M_n$ be the space of $n \times n$ real matrices, and consider the following equivalence relation on $M_n$:
$A \sim B$ if there exist $Q \in O(n)$ such that $A=QB$.
Can we characterise nicely the equivalence classes of this relation?
By comparison, the usual "orthogonal equivalence", i.e. $A \sim B \iff A=QBU$ for some $Q,U \in O(n)$ gives rise to the notion of SVD-i.e. each equivalence class corresponds to a specific finite sequence of singular values. Thus, the singular values are the "invariants" which classify the equivalence classes.
Is there any sensible way to associate a list of classifying invariants to the new relation in a similar way?
Of course, we shall need "more invariants" to distinguish between different classes: It will be something like "the same singular values+something else".
In particular, we have $A \sim B \implies \ker A=\ker B$, but for invertible matrices this does not really add any information.
Even finding some non-trivial sufficient conditions for when $A \sim B$ would be interesting: Having the same singular values is certainly necessary, but is far from sufficient, which can be checked directly even at dimension $2$. ($\Sigma$ and $\Sigma Q$ are not equivalent generically).