Galam reshuffling introduced in opinion dynamics models is investigated under the nearest neighbor Ising model on a square lattice using Monte Carlo simulations. While the corresponding Galam analytical critical temperature T_C approx 3.09 [J/k_B] is recovered almost exactly, it is proved to be different from both values, not reshuffled (T_C=2/arcsinh(1) approx 2.27 [J/k_B]) and mean-field (T_C=4 [J/k_B]). On this basis, gradual reshuffling is studied as function of 0 leq p leq 1 where p measures the probability of spin reshuffling after each Monte Carlo step. The variation of T_C as function of p is obtained and exhibits a non-linear behavior. The simplest Solomon network realization is noted to reproduce Galam p=1 result. Similarly to the critical temperature, critical exponents are found to differ from both, the classical Ising case and the mean-field values.
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