What's the opposite of a cross product?

As Fabian wrote, $b$ is not uniquely determined by $a$ and $c$. Moreover, there is no solution unless $a$ and $c$ are orthogonal. If $a$ and $c$ are orthogonal, then the solutions are $(c \times a)/(a\, . a) + t a$ for arbitrary scalars $t$.

The name "product" for the cross product is unfortunate. It really should not be thought of as a product in the ordinary sense; for example, it is not even associative. Thus one should not expect it to have properties analogous to the properties of ordinary multiplication.

What the cross product really is is a Lie bracket.

Alright, let's look for an inverse here:

The cross product is restricted to $\mathbb{R}^3$ (assuming reals--but that isn't really a constraint here, I don't think). So we have:

$$ \langle x_1, y_1, z_1\rangle\times\langle x_2, y_2, z_2\rangle = \langle y_1z_2 - z_1y_2, z_1x_2 - x_1z_2, x_1y_2 - y_1x_2\rangle $$

Then we have:

\begin{align} \langle x_3, y_3, z_3\rangle\times\left(\langle x_1, y_1, z_1\rangle\times\langle x_2, y_2, z_2\rangle\right) = & \langle y_3(x_1y_2 - y_1x_2) - z_3(z_1x_2 - x_1z_2),\\ &z_3(y_1z_2 - z_1y_2) - x_3(x_1y_2 - y_1x_2),\\ & x_3(z_1x_2 - x_1z_2) - y_3(y_1z_2 - z_1y_2)\rangle \end{align}

Now this looks very complicated (and it is--if you need to solve it "as is"). Instead we can imagine that we already know $\vec{n}' = \vec{v}_1\times\vec{v_2}$. Then this becomes:

\begin{align} \langle x_3, y_3, z_3\rangle\times\vec{n}' = & \langle y_3n'_z - z_3n'_y, z_3n'_x - x_3n'_z, x_3n'_y - y_3n'_x\rangle \end{align}

Now we are trying to find $\langle x_3, y_3, z_3\rangle$ which is the inverse. So we set this equal to the second argument of the original cross product, we have a set of linear equations for three unknowns ($x_3, y_3, z_3$):

$$ x_2 = y_3n'_z - z_3n'_y \\ y_2 = z_3n'_x - x_3n'_z \\ z_2 = x_3n'_y - y_3n'_x $$

The matrix gives:

$$ \begin{pmatrix} 0 & n_z' & -n_y' \\ -n_z' & 0 &n_x' \\ n_y' & -n_x' & 0 \end{pmatrix} \times \begin{pmatrix}x_3 \\ y_3 \\ z_3\end{pmatrix} = \begin{pmatrix}x_2 \\ y_2 \\ z_2\end{pmatrix} $$

There is no solution when the determinate of this matrix is zero--which is always the case: $n_z'x_x'n_y'- n_y'n_z'n_x' = 0$. This means that unless $\langle x_2, y_2, z_2\rangle$ is $\vec{0}$ there is no solution (since a matrix that has a determinant equal to zero only has a solution when the RHS is zero--in which case it has infinite solutions).

Edit (in regards to comments)

While it's not strictly true that just because the RHS is non-zero (and the matrix is degenerate) there won't be a solution, it is true "in general". Meaning that you can only find solutions in very special cases. This still suggests that there is not an inverse to the cross product (except in very special cases).