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The effect of interactions on the saturation remanence of assemblies of identical, uniaxially anisotropic, single-domain particles is calculated using a spatial-mean interaction field. The particle easy-axis directions are assumed known and given by a distribution function. The remanence is determined by finding the magnetization orientation functional of the particle easy-axis orientation which minimizes the total assembly energy. Curves of remanence versus interaction strength (assembly packing fraction) are shown for a) randomly oriented assemblies of spherical particles with uniaxial crystalline anisotropy only, b) randomly oriented assemblies of acicular particles with shape anisotropy, and c) oriented assemblies of acicular particles with 6:1 aspect ratio. Disregarding external sample-shape demagnetization effects, this model always yields increased remanences due to interactions. A criterion is given which predicts when external shape effects are capable of dominating the net interaction field to yield a reduction in remanence. The applicability of these results and extensions of the theory are discussed in relation to particle assemblies in magnetic tape and high coercivity CoNiP films.