How to compare or find common properties of unitary matrices?

Question

I have multiple $3\times 3$ unitary matrices, all of which optimize a function $f$. I want to find the structure or common pattern/properties of these unitaries.

The function $f$ to be optimized has 9 parameters(real numbers). These 9 parameters are used to generate a $3\times 3$ unitary matrix, and then use it to maximize a function. The function works like:

1. Generate a unitary matrix $U$ from the 9 input parameters.
2. M = I $ \otimes $ I $ \otimes $ U, where is $I$ is the $3\times 3$ identity matrix
3. E = MQ, where $M$ is a $27 \times 27$ matrix and $Q$ is $27 \times 1$ matrix
4. Reshape $E$ to be of size $3 \times 9$
5. Compute SVD of $E$ and let $s_1,s_2,s_3$ be the singular values of E.
6. Maximise $-\sum_{x_i} p(x_i)\ log \ p(x_i)$, where $x = \{s_1^2, s_2^2, s_3^2\}$

I have found the parameters that maximize this function. The parameters are not unique. Hence, I have many different unitary matrices, all of which maximise $f$. And I don't know if there is anything common about these unitaries.

So far I have tried comparing their eigenvalues, eigenvectors, trace, determinant and inverse. But none of these values are equal.

What other ways can I try to find the common structure of these unitaries?