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\*}



Correspondence between $SBT (n)$ and $W(B_n)$
Before getting into my answer, let me explain what I understand by a bitableau of size $n$ (I hope it is the same as what you mean). This is a pair of tableaux $(P, P')$, where each of the integers $1,\dotsc,n$ occurs exactly once, and moreover, the numbers increase along rows and columns in both $P$ and $P'$. The shape of the bitableaux is $(\mu, \mu')$ where $\mu$ and $\mu'$ are the shapes of $P$ and $P'$. Bitableaux of shape $(\mu, \mu')$ correspond to maximal chains in the Bratelli diagram of the sequence of groups $W(B\_n)$ which begin at $(\emptyset, \emptyset)$ and end at $(\mu, \mu')$.


The Robinson-Schensted correspondence has a generalization - the RSK correspondence. This correspondence is between (nonnegetive) integer matrices and pairs of semistandard Young tableaux of the same shape (see Knuth's 1970 paper in PJM, or [my book](http://www.imsc.res.in/~amri/rtcv/)). If the rows of the integer matrix sum to $\lambda = (\lambda\_1,\dots, \lambda\_l)$, and the columns sum to $\mu = (\mu\_1,\dotsc, \mu\_m)$, then the correspondence associates to $A$ a pair $(P, Q)$ of semistandard young tableaux, $P$ having content $\lambda$, and $Q$ having content $\mu$.


The matrix $A$ is said to be a partial permutation if all the row and column sums are at most $1$. Now, an element $w\in W(B\_n)$ may be regarded as a pair of partial permutations, $(A, B)$ such that $A + B$ is a permutation matrix as follows: $A\_{ij} = 1$ if $w:j\mapsto i$, and is $0$ otherwise. $B\_{ij} = 1$ if $w:j\mapsto -i$, and is $0$ otherwise.


Now apply the RSK correspondence to $A$ and to $B$, getting $(P\_A, Q\_A)$ and $(P\_B, Q\_B)$. Then $(P\_A, P\_B)$ and $(Q\_A, Q\_B)$ are both bitableaux of the same shape.


Consider for example, the element (in your notation)
$$w = \begin{pmatrix} 1 & 2 & 3 & -1 & -2 & -3\\
2 & -1 & -3 & -2 & 1 & 3 \end{pmatrix}$$
corresponds to the matrices:
$$
A = \begin{pmatrix}
0 & 0 & 0\\
1 & 0 & 0\\
0 & 0 & 0\end{pmatrix}, \quad 
B = \begin{pmatrix}
0 & 1 & 0\\
0 & 0 & 0\\
0 & 0 & 1\end{pmatrix}
$$
Under the RSK correspondence $A$ maps to $P=(1)$, $Q = (2)$ (of shape $(1)$), while $B$ corresponds to $(2, 3)$ and $(1, 3)$ (of shape $(2)$). So the bitableaux pair for $w$ is 
$(1), (2, 3)$ and $(2), (1, 3)$.


The relation to domino tableaux is rather interesting. Given a bipartition $(\alpha, \beta)$, let $\lambda$ denote the unique partition whose $2$-core is trivial and whose $2$-quotient is $(\alpha, \beta)$.
Then adding a box to either $\alpha$ or $\beta$ corresponds to adding a domino to $\lambda$ (see the book by James and Kerber, or the paper of Frame, Robinson and Thrall for details). The size of $\lambda$ will be twice the sum of the sizes of $\alpha$ and $\beta$. Thus we may regard a bitableau of size $n$ as a domino-tableau of size $2n$. Thus the correspondence that I described above may also be regarded as a bijection from $W(B\_n)$ onto the set of pairs of domino tableaux of the same shape.


In the running example, the associated domino tableau are $P = \begin{array}{cccc}1&2&3&3\\1&2&&\end{array}$ and $Q = \begin{array}{cccc}1&1&3&3\\2&2&&\end{array}$.


Here is some Sage code for the computation:



```
from sage.combinat.tableau import from_chain

def matrices(w):
    n = len(w)
    A = Matrix(ZZ, n, n, lambda i,j: int(w[j] == i+1))
    B = Matrix(ZZ, n, n, lambda i,j: int(w[j] == -i-1))
    return A, B

def domino_rsk(w, dominoes=True):
    n = len(w)
    A, B = matrices(w)
    P1, Q1 = tuple(RSK(A))
    P2, Q2 = tuple(RSK(B))
    if dominoes:
        ch_P1 = P1.to_chain(max_entry=n)
        ch_P2 = P2.to_chain(max_entry=n)
        ch_Q1 = Q1.to_chain(max_entry=n)
        ch_Q2 = Q2.to_chain(max_entry=n)
        P = from_chain([Partition(core=[], quotient=[ch_P1[i], ch_P2[i]]) for i in range(n+1)])
        Q = from_chain([Partition(core=[], quotient=[ch_Q1[i], ch_Q2[i]]) for i in range(n+1)])
        return P, Q
    else:
        return [P1,P2], [Q1, Q2]

```

For the elements of $W(B\_2)$, I got (in terms of bitableaux):



```
[1, 2] ([[[1, 2]], []], [[[1, 2]], []])
[1, -2] ([[[1]], [[2]]], [[[1]], [[2]]])
[-1, 2] ([[[2]], [[1]]], [[[2]], [[1]]])
[-1, -2] ([[], [[1, 2]]], [[], [[1, 2]]])
[2, 1] ([[[1], [2]], []], [[[1], [2]], []])
[2, -1] ([[[1]], [[2]]], [[[2]], [[1]]])
[-2, 1] ([[[2]], [[1]]], [[[1]], [[2]]])
[-2, -1] ([[], [[1], [2]]], [[], [[1], [2]]])

```

and in terms of domino tableaux:



```
[1, 2] ([[1, 2, 2], [1]], [[1, 2, 2], [1]])
[1, -2] ([[1, 2], [1, 2]], [[1, 2], [1, 2]])
[-1, 2] ([[1, 1], [2, 2]], [[1, 1], [2, 2]])
[-1, -2] ([[1, 1, 2, 2]], [[1, 1, 2, 2]])
[2, 1] ([[1], [1], [2], [2]], [[1], [1], [2], [2]])
[2, -1] ([[1, 2], [1, 2]], [[1, 1], [2, 2]])
[-2, 1] ([[1, 1], [2, 2]], [[1, 2], [1, 2]])
[-2, -1] ([[1, 1], [2], [2]], [[1, 1], [2], [2]])

```

This is not the same as the [answer obtained by Brian Hopkins](https://mathoverflow.net/a/251810/9672), whose dominoes have an extra $0$, and I must admit that I have not looked carefully the references he gave as yet (and am not convinced that this question should need one to read four papers in *Compositio*). However, if I tweak my code so that $\lambda$ has core $(1)$ and quotient $(\alpha,\beta)$, then I do seem to get his answer. 


