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When one performs a non-commutative operation on two numbers there is usually a simple way of swapping between the two possible answers, for example, $5-4=1$ and $4-5=-1$; one can be mapped onto the other simply by multiplying by $-1$

This also holds for division: $1/5=0.2$ can be mapped onto $5/1=5$ by calculating the answer's multiplicative inverse (raising it to $-1$, pattern perhaps?).

Do matrices have any equivalent function that would map $\textbf{AB}$ onto $\textbf{BA}$ without having to know $\textbf{A}$ or $\textbf{B}$?

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    No. For instance you can have $AB=0$ and $BA\ne0$. – 2017-02-02
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    @egreg Thanks for answering! Could you post that as an answer so that I can accept it? – 2017-02-03

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There is no generally valid connection between $AB$ and $BA$, even when $A$ and $B$ are square matrices. As an instance, consider $$ \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} $$ whereas $$ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$ You see that you cannot even predict the rank of the reversed product. More complicated examples can be built with higher order matrices.

The only case when you can certainly predict $BA$ from $AB$, for square matrices, is when $AB=I$ (the identity matrix), because in this case $BA=I$ as well.