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A strategy to improve the performance and reduce the memory footprint of simulations on meshes with spatial reflection symmetries is presented in this work. By using an appropriate mirrored ordering of the unknowns, discrete partial differential operators are represented by matrices with a regular block structure that allows replacing the standard sparse matrix–vector product with a specialised version of the sparse matrix-matrix product, which has a significantly higher arithmetic intensity. Consequently, matrix multiplications are accelerated, whereas their memory footprint is reduced, making massive simulations more affordable. As an example of practical application, we consider the numerical simulation of turbulent incompressible flows using a low-dissipation discretisation on unstructured collocated grids. All the required matrices are classified into three sparsity patterns that correspond to the discrete Laplacian, gradient, and divergence operators. Therefore, the above-mentioned benefits of exploiting spatial reflection symmetries are tested for these three matrices on both CPU and GPU, showing up to 5.0x speed-ups and 8.0x memory savings. Finally, a roofline performance analysis of the symmetry-aware sparse matrix–vector product is presented. • Strategy to accelerate CFD simulations on meshes with spatial reflection symmetries. • Replacement of SpMV with a specialised version of the more-compute intensive SpMM • Implementation of a lighter sparse matrix storage format accounting for symmetries. • Hierarchical multilevel MPI+OpenMP+OpenCL/CUDA parallelisation. • Numerical tests on CPUs and GPUs show up to 5x speed-ups and 8x memory savings. [ABSTRACT FROM AUTHOR]