Let $p$ is a product of two non-zero ordinal numbers. Knowing $p$ we can restore each of these two ordinals, right?
How can we denote each of these two ordinals (supposing we know the value of $p$)? I suspect, these may be called "projections", right?
No, you can’t always reconstruct $\alpha$ and $\beta$ from $\alpha\cdot\beta$. For example, $n\cdot\omega=\omega$ for every non-zero finite ordinal $n$.