},{\varepsilon}}^{n-1}$ interpolates the commutative sphere $S_{{ {\mathbb R} }}^{n-1}$ and the free sphere $S_{{ {\mathbb R} },+}^{n-1}$, this is *not* the case for the ${\varepsilon}$-orthogonal quantum groups – if ${\varepsilon}\neq {\varepsilon}_{\operatorname{comm}}$ and ${\varepsilon}\neq {\varepsilon}_{\operatorname{free}}$, we have (see Prop. \[PropNoInterpolation\]): $$O_n\not\subset O_n^{\varepsilon}\subsetneq O_n^+\qquad\textnormal{and}\qquad S_n\not\subset O_n^{\varepsilon}\subsetneq O_n^+$$

Our original question about the symmetries of ${\varepsilon}$-independence is answered by the following weak version of a de Finetti theorem.

Let $x_1,\ldots,x_n$ be selfadjoint random variables in a noncommutative probability space $(A,{\varphi})$ such that $x_ix_j=x_jx_i$ if ${\varepsilon}_{ij}=1$. If $x_1,\ldots,x_n$ are ${\varepsilon}$-independent and identically distributed, then their distribution is invariant under $S_n^{\varepsilon}$.

We extend this de Finetti Theorem to the invariances by the quotients of $H_n^+$, $B_n^+$ and $O_n^+$ by $\mathring R^{\varepsilon}$.

Preliminaries on ${\varepsilon}$-independence
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The notion of ${\varepsilon}$-independence as a mixture of classical and free independence has been introduced by the first author and Wysoczanski [@SpW] very recently, rediscovering Mlotkowski’s $\Lambda$-independence [@Mlo]. We review its main features here. Throughout the whole article we denote by ${\varepsilon}$ an $n\times n$-matrix such that:

-   ${\varepsilon}_{ij}\in\{0,1\}$ for all $i,j=1,\ldots,n$

-   ${\varepsilon}$ is symmetric

-   ${\varepsilon}_{ii}=0$ for all $i=1,\ldots,n$

\[DefEpsInd\] Let $(A,{\varphi})$ be a noncommutative probability space. We say that unital subalgebras $A_1,\ldots,A_n\subset A$ are *${\varepsilon}$-independent*, if we 