We determine couple-stress moduli and characteristic lengths of a two-dimensional matrix-inclusion composite, with inclusions arranged in a periodic square array and both constituents linear elastic of Cauchy type. In the analysis we replace this composite by a homogeneous planar, orthotropic, couple-stress continuum. A generalization of the original Mindlin's (1963) derivation of field equations for such a continuum results in two (not just one!) characteristic lengths. We evaluate the couplestress properties from the response of a unit cell under several types of boundary conditions: displacement, displacementperiodic, periodic and mixed, and traction controlled. In the parametric study we vary the stiffness ratio of both phases to cover a range of different media from nearly porous materials through composites with very stiff inclusions. We find that the aforementioned boundary conditions result in hierarchies of orthotropic couple-stress moduli, whereas both characteristic lengths are fairly insensitive to boundary conditions, and fall between 0.12% and 0.22% of the unit cell size for the inclusions' volume fraction of 18%. 2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.