Assume $\langle A,B \rangle = 0$, where $A,B$ are $n \times n$ matrices and $\langle , \rangle$ is the Frobenius inner product. (Also known as the Hilbert–Schmidt inner product).
What does it mean intuitively? In particular, is there some relevant interpretation when we think of $A,B$ as linear transformations?
What about when $\langle A,B \rangle > 0$?
I wish I had a better answer for this, and I'm interested to hear what other people say. But one thing to note is that the trace has a physical interpretation: $$\operatorname{Tr}A = \left. \frac{d}{dt}\right|_{t=0}\det(I + tA) = \left. \frac{d}{dt}\right|_{t=0}\det (e^{tA}).$$
Therefore if you flow along the vector field $F(x)=Ax$, then $\operatorname{Tr}A$ is the amount that this flow distorts volume. (This is paraphrased from a Stack answer somewhere whose link I have lost.)
So to say that $\operatorname{Tr}B^*A=0$ is to say that if you flow by $y'(t) = B^*Ay(t)$ you are not changing volume. (This is the reason why the traceless matrices are the tangent space to $SL_n(\mathbb{R})$ at the identity, since $SL_n(\mathbb{R})$ is defined by its constant determinant.)
Sorry I don't have anything better.