Focal-plane speckles set important sensitivity limits on ground- or space-based imagers and coronagraphs that may be used to search for faint companions, perhaps ultimately including exoplanets, around stars. As speckles vary with atmospheric fluctuations or with drifting beamtrain aberrations, they contribute speckle noise proportional to their full amplitude. Schemes to suppress speckles are thus of great interest. At high adaptive correction, speckles organize into species, represented by algebraic terms in the expansion of the phase exponential, that have distinct spatial symmetry, even or odd, under spatial inversion. Filtering speckle patterns by symmetry may eliminate a disproportionate fraction of the speckle noise while blocking (only) half of the image signal from the off-axis companion being sought. The fraction of speckle power and hence of speckle noise in each term will vary with degree of correction, and so also will the net symmetry in the speckle pattern. Systematic numerical investigations are presented of the net symmetry of noise variance as a function of adaptive correction, i.e., Strehl ratio S, and deformable mirror actuator density D/a, where a is the deformable mirror actuator spacing referred to the pupil of diameter D, which controls the characteristic transverse spatial frequency of the wavefront. The degree of speckle symmetry is found to be substantial even at current relatively modest ground-based corrections (S=0.6, D/a=16 in the near-infrared). With parameters representative of "extreme" adaptive optics of the near future (S=0.99, D/a=100), the antisymmetric noise variance fraction is 0.99967 averaged over two Airy rings in the inner halo, so simple image processing (symmetry filtering) can improve the net speckle-noise-limited companion-detection SNR by a factor of about 28. Analogous processing can enhance SNR in coronagraphic searches, where speckle patterns before processing are predominantly symmetric.