Let $X$ and $Y$ be independent random symmetric matrices. What can one say about $\mathbb{E} [X Y X Y]$ or $\mathrm{trace} \mathbb{E} [X Y X Y]$ in terms of properties of $X$ and $Y$?
In particular, can we compute such expectations if $X$ is Wishart distributed and $Y$ is inverse-Wishart distributed (i.e. the inverse of $Y$ is Wishart distributed)?
You could compute these expectation values from the known marginal distribution of the matrix elements of the Wishart and inverse Wishart ensembles; as a simpler test case here is the answer for a single product $XY$ of a $(p,n)$ Wishart and a $(p,\nu)$ inverse Wishart matrix (with identity scale matrices): $${\rm tr}\,\mathbb{E}[XY]=\sum_{k_1,k_2=1}^{p}\mathbb{E}[X_{k_1k_2}]\mathbb{E}[Y_{k_2k_1}]=\frac{pn}{\nu-p-1}.$$
This falls within the domain of free probability, at least when the dimension is large. That is, if the matrices are say unitarily invariant and independent, and if we denote by $tr_N$ the trace divided by the dimension, and writing $\bar x=tr_N X$, $\bar y=tr_N Y$, you get $$E tr_N((X-\bar xI_N)(Y-\bar yI_N)(X-xI_N)(Y-yI_N))\to_{N\to\infty} 0.$$ You also have (from the same reasoning) that $Etr_N XY-\bar x\bar y\to 0$ and that $Etr_N XYX=Etr_N(X^2Y)= tr_N(X^2) \bar y+o(1)$, you get $$Etr_N(XYXY)=o(1)+2\bar x^2 tr_N Y^2+2\bar y^2 tr_N X^2-3\bar x^2 \bar y^2$$
