It is shown that a wide-sense stationary process is strictly stationary (i) only if it is rotation invariant, (ii) if it is rotation invariant and has a (two dimensional) random spectral measure with independent increments.
A. Larry Wright. "On a Theorem of Maruyama." Ann. Probab. 8 (4) 851 - 852, August, 1980. https://doi.org/10.1214/aop/1176994671