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ACo₂As₂ (A = Ca, Sr, Ba) are tetragonal itinerant magnets that display a dichotomy typical of their As-As hybridization-driven magnetoelastic interactions: CaCo₂As₂ has A-type antiferromagnetic (AFM) order with Co ordered moments aligned along c, and SrCo₂As₂ and BaCo₂As₂ do not order. Ca_1-xSr_xCo₂As₂ shows a series of magnetic phases, with transitions between them at x ≈ 0.2, 0.3, and 0.5. Magnetization data show A-type order as in CaCo₂As₂ for x < 0.2 and no magnetic order for x > 0.5. Here, we present and discuss quantitative results from neutron and high-energy x-ray diffraction experiments on single crystals with x = 0.25 and 0.40. Rather than A-type order, we find extremely weak magnetic Bragg peaks corresponding to a propagation vector of (0, 0, 0.5) for both samples. The ordered moment lies along c for x = 0.25 but in the ab plane for x = 0.40. No structural phase transitions accompany the AFM transitions.