Say we play a game where we pick a random point $\rho \in \mathcal{B}(\mathcal{H})$. The rules are, you can pick any 4 unitary $(\mathcal{U}:\mathcal{B}(\mathcal{H}) \rightarrow \mathcal{B}(\mathcal{H}))$ maps $\mathcal{U}_1, \mathcal{U}_2, \mathcal{U}_3$, and $\mathcal{U}_4$ to apply onto the state, such that their joint action amounts to the identity:
$(\mathcal{U}_4 \circ \mathcal{U}_3 \circ \mathcal{U}_2 \circ \mathcal{U}_1)[\rho] = \mathcal{I}[\rho]=\rho$,
and $\mathcal{U}_i \mathcal{U}_i^{\dagger} = \mathcal{U}_i^{\dagger} \mathcal{U}_i = \mathcal{I}$.
Is there an elegant way to find all the "strategies" which make this work? Or alternatively, what's the general relationship between these 4 maps such that the above is true.
My thoughts
By the power of associativity and the unitary condition above, there's a few obvious strategies which work.
Pick three random rotations $\mathcal{U}_1, \mathcal{U}_2, \mathcal{U}_3$, and make the 4th the $\dagger$ of the first three: $\mathcal{U}_4 = (\mathcal{U}_1 \circ \mathcal{U}_2 \circ \mathcal{U}_3)^{\dagger}$. The same works if the first unitary is the $\dagger$ of the last three too.
Pick two random rotations and make the remaining two be the $\dagger$'s of each of them e.g. $\mathcal{U}_2 = \mathcal{U}_1^{\dagger}$ and $\mathcal{U}_4 = \mathcal{U}_3^{\dagger}$. This works in any order.
Pick 4 unitaries that amount to a $2\pi$ rotation in a plane.
Those are the obvious ones. But what about a general statement where this is true? Is there a framework where this question becomes simple to answer? The tags show my guesses, but I'd be open to anything.