Suppose I have square unitary matrices $A, B, C, D$. I also have the unitary matrix $X = ABCD$ (or more generally, $X = X_1X_2\cdots X_n$). Is there a way to recover $Y=DCBA$ (or more generally, $Y = X_n\cdots X_2 X_1$) only using operations on $X$?
I'm also interested in knowing how much incremental information about $X$ is needed before we learn something about $Y$. Specifically, if we only know the first column of $X$, can we say anything about what $Y$ looks like?
On
I see a basic "obstruction" that begins with the product of two matrices $AB$:
As $X=AB$ is identical to $(AC)(C^{-1}B)$ for any invertible $C$, being able to retrieve the product $Y=BA$ in a unique way from $X$ is deemed to fail because it would mean in particular that $Y=(C^{-1}B)(AC)=C^{-1}(BA)C$, which is similar but in general non identical to $BA$.
No. $DCBA$ is not a function of $ABCD$. Let $A$ and $B$ be any two non-commuting unitary matrices, such as $$ A=\pmatrix{0&-1\\ 1&0},\ B=\pmatrix{1&0\\ 0&-1}. $$ Now let $(A_1,B_1,C_1,D_1)=(A,B,I,I)$ and $(A_2,B_2,C_2,D_2)=(AB,I,I,I)$. Then $A_1B_1C_1D_1=AB=A_2B_2C_2D_2$ but $D_1C_1B_1A_1=BA\ne AB=D_2C_2B_2A_2$.