The popular NL-means denoising algorithm proposes to modify the intensity of each voxel of an image by a weighted sum of the intensities of similar voxels. The success of the NL-means rests on the fact that there are typically enough such similar voxels in natural, and even medical images; in other words, that there is some self-similarity/redundancy in such images. However, similarity between voxels (or rather, between patches around them) is usually only assessed in a spatial neighbourhood of the voxel under study. As the human brain exhibits approximate bilateral symmetry, one could wonder whether a voxel in a brain image could be more accurately denoised using information from both ipsi- and contralateral hemispheres. This is the idea we investigate in this paper. We define and compute a mid-sagittal plane which best superposes the brain with itself when mirrored about the plane. Then we use this plane to double the size of the neighbourhoods and hopefully find additional interesting voxels to be included in the weighted sum. We evaluate this strategy using an extensive set of experiments on both simulated and real datasets.