be a symmetric matrix with ${\varepsilon}_{ij}\in\{0,1\}$ and ${\varepsilon}_{ii}=0$. If variables $x_1,\ldots,x_n$ are ${\varepsilon}$-independent, then:

-   $x_i$ and $x_j$ are free in the case ${\varepsilon}_{ij}=0$

-   and $x_i$ and $x_j$ are independent in the case ${\varepsilon}_{ij}=1$ (in particular $x_ix_j=x_jx_i$ in this situation).

If all entries of ${\varepsilon}$ are zero (${\varepsilon}={\varepsilon}_{\operatorname{free}}$), we obtain Voiculescu’s free independence; if all non-diagonal entries of ${\varepsilon}$ are one (${\varepsilon}={\varepsilon}_{\operatorname{comm}}$), we obtain classical independence.

It is well-known that independences can be characterized via distributional symmetries with the help of de Finetti type theorems. For instance, classical independence is equivalent to invariance under actions of the symmetric group $S_n$, whereas free independence is characterized [@SpK] by invariance under actions of Wang’s [@WangSym] quantum symmetric group $S_n^+$. Our aim is to find the right quantum groups corresponding to the above ${\varepsilon}$-independence.

For doing so, we first define noncommutative ${\varepsilon}$-spheres $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$ by: $$C(S_{{ {\mathbb R} },{\varepsilon}}^{n-1}):=C^*(x_1,\ldots,x_n\;|\; x_i=x_i^*, \sum_i x_i^2=1, x_ix_j=x_jx_i, \textnormal{ if } {\varepsilon}_{ij}=1)$$ For ${\varepsilon}={\varepsilon}_{\operatorname{comm}}$ the above $C^*$-algebra is commutative and it is nothing but the algebra of continuous functions over the real sphere $S_{{ {\mathbb R} }}^{n-1}\subset{ {\mathbb R} }^n$. For ${\varepsilon}={\varepsilon}_{\operatorname{free}}$ we obtain Banica and Goswami’s [@BanGos] free version $S^{n-1}_{{ {\mathbb R} },+}$ of 