  We introduce and solve exactly a family of invariant 2x2 random matrices,
depending on one parameter \eta, and we show that rotational invariance and
real Dyson index \beta are not incompatible properties. The probability density
for the entries contains a weight function and a multiple trace-trace
interaction term, which corresponds to the representation of the
Vandermonde-squared coupling on the basis of power sums. As a result, the
effective Dyson index \beta_{eff} of the ensemble can take any real value in an
interval. Two weight functions (Gaussian and non-Gaussian) are explored in
detail and the connections with \beta-ensembles of Dumitriu-Edelman and the
so-called Poisson-Wigner crossover for the level spacing are respectively
highlighted. A curious spectral twinning between ensembles of different
symmetry classes is unveiled. The proposed technical tool more generically
allows for designing actual matrix models which i) are rotationally invariant;
ii) have a real Dyson index \beta_{eff}; iii) have a pre-assigned confining
potential or alternatively level-spacing profile. The analytical results have
been checked through numerical simulations with an excellent agreement.
Eventually, we discuss possible generalizations and further directions of
research.
