Stochastic dynamical systems with continuous symmetries arise commonly in nature and often give rise to coherent spatio-temporal patterns. However, because of their random locations, these patterns are not captured well by current order reduction techniques, and a large number of modes is typically necessary for an accurate solution. In this work, we introduce a new methodology for efficient order reduction of such systems by combining (i) the method of slices [C. W. Rowley and J. E. Marsden, Phys. D, 142 (2000), pp. 1-19; S. Froehlich and P. Cvitanovi'c, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), pp. 2074-2084], a symmetry reduction tool, and (ii) any standard order reduction technique, resulting in efficient mixed symmetry-dimensionality reduction schemes. In particular, using the dynamically orthogonal (DO) equations [T. P. Sapsis and P. F. J. Lermusiaux, Phys. D, 238 (2009), pp. 2347-2360] in the second step, we obtain a novel nonlinear symmetry-reduced dynamically orthogonal (SDO) scheme. We demonstrate the performance of the SDO scheme on stochastic solutions of the one-dimensional Korteweg-de Vries and two-dimensional Navier-Stokes equations. Keywords: model order reduction, stochastic dynamical systems, symmetry reduction;
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