<s>[INST] Correspondence between $SBT (n)$ and $W(B_n)$ Let $W(B_n)$ be a Weyl group of type $B_n$ and $SBT (n)$ the set of standard bitableaux of size $n$. Similar to Robinson-Schensted correspondences, I know that there exists a map $W(B_n) \to SBT (n) \times SBT (n)$ such that the image of $W(B_n)$ is the set of the same shape pairs of standard bitableaux. 
But, I don't know what the map $W(B_n) \to SBT (n) \times SBT (n)$ is. In particular, $W(B_2) \to SBT (2) \times SBT (2)$? Here
\begin{align*}
W(B_2) \cong & \left\{ 
\begin{pmatrix} 
1 & 2 &  -1 & -2 \\
1 & 2 &  -1 & -2 
\end{pmatrix}, \begin{pmatrix}
1 & 2 &  -1 & -2 \\
2 & 1 &  -2 & -1
\end{pmatrix}, \begin{pmatrix}
1 & 2 &  -1 & -2 \\
-1 & 2 &  1 & -2
\end{pmatrix}, \right. \\
& \left. \begin{pmatrix}
1 & 2 &  -1 & -2 \\
1 & -2 &  -1 & 2
\end{pmatrix}, 
\begin{pmatrix}
1 & 2 &  -1 & -2 \\
-2 & 1 &  2 & -1
\end{pmatrix}, \begin{pmatrix}
1 & 2 &  -1 & -2 \\
2 & -1 &  -2 & 1
\end{pmatrix}, \right. \\
& \left. \begin{pmatrix}
1 & 2 &  -1 & -2 \\
-1 & -2 & 1 & 2
\end{pmatrix}, \begin{pmatrix}
1 & 2 &  -1 & -2 \\
-2 & -1 &  2 & 1
\end{pmatrix}\right\}.
\end{align*}
 [/INST] Devra Garfinkle developed such a correspondence to pairs of domino tableaux in a series of Compositio Mathematica papers in the early 1990s.  (The first & third, but curiously not the second, are available through EUDML: https://eudml.org/doc/90031 and https://eudml.org/doc/90244.)  More succinct summaries are given in the work of McGovern, van Leeuwen, Shimizono, Pietraho, Taskin, etc.
The $W(B_2)$ example you request doesn't involve any of the horizontal/vertical domino rotations that make this theory tricky, just a little bumping.  Here are the 8 domino tableaux pairs in the order you gave the $W(B_2)$ elements.  Note that all but the 5th and 6th are involutions, so their left and right tableaux are the same in all but those.  [If someone knows how to TeX these into nice domino pictures, please do so and let me know how you do it.]
\begin{gather*}
\left(\begin{matrix} 0 & 1 & 1 & 2 & 2 \end{matrix},  \quad \begin{matrix} 0 & 1 & 1 & 2 & 2 \end{matrix}\right), \quad
\left(\begin{matrix} 0 & 1 & 1 \\ 2 & 2 \end{matrix},  \quad \begin{matrix} 0 & 1 & 1 \\ 2 & 2 \end{matrix}\right), \\ \\
\left(\begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix},  \quad \begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix}\right), \quad
\left(\begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix},  \quad \begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix}\right), \\ \\
\left(\begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix},  \quad \begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix}\right), \quad
\left(\begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix},  \quad \begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix}\right), \\ \\
\left(\begin{matrix} 0 & 2 \\ 1 & 2 \\ 1 \end{matrix},  \quad \begin{matrix} 0 & 2 \\ 1 & 2 \\ 1 \end{matrix}\right), \quad
\left(\begin{matrix} 0 \\ 1 \\ 1 \\ 2 \\ 2 \end{matrix},  \quad \begin{matrix} 0 \\ 1 \\ 1 \\ 2 \\ 2 \end{matrix}\right).
\end{gather*}
(The domino tableaux for $W(C_2)$ also have dominoes labeled 1 and 2, but no 0 square.)
</s>