What is the motivation behind symplectic/orthogonal content? Here $\lambda'$ is the conjugate partition of $\lambda=(\lambda_1,\lambda_2,\dots)$ and cells are in the Young diagram.
The symplectic content of cell $(i,j)$ of $\lambda$ is defined by
$$c_{sp}(i,j)=\begin{cases} \lambda_i+\lambda_j-i-j+2 \qquad \text{if $i>j$} \\
i+j-\lambda_i'-\lambda_j' \qquad \qquad \text{if $i\leq j$}.\end{cases}$$
The orthogonal content of cell $(i,j)$ of $\lambda$ is defined by
$$c_{O}(i,j)=\begin{cases} \lambda_i+\lambda_j-i-j \qquad \qquad \text{if $i\geq j$} \\
i+j-\lambda_i'-\lambda_j'-2 \qquad  \text{if $i< j$}.\end{cases}$$
Although I have used these in my analysis, I still wonder:

QUESTION. What is the motivation behind these definition choices for the "contents"?

 A: As I guessed in a comment above, there is apparently a symplectic/orthogonal hook-content formula which uses these notions. See "Hook-content Formulae for Symplectic and Orthogonal Tableaux" by Campbell and Stokke https://doi.org/10.4153/CMB-2011-105-7.
Warning: they seem to have withdrawn the arXiv version of their paper (https://arxiv.org/abs/0710.4155) for reasons that are not clear to me.
EDIT: Regarding priority, I see Campbell and Stokke cite El Samra and King for the formula for the evaluation of the symplectic/orthogonal characters at $(1,1,\ldots,1)$ (i.e., pure counting of tableaux); what they do that is new is the principal evaluation $(1,q,\ldots,q^{n-1})$ (i.e., $q$-counting). Perhaps the arXiv withdrawal is about what Per said: copyright.
A: A hook-content formula, using the contents $c_{sp}(i,j)$ and $c_O(i,j)$,  for the dimensions of the irreducible polynomial representations of the symplectic and orthogonal groups, goes back to Ron King. I believe the relevant paper is https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0008414X00053086, but I did not check for sure.
