Abstract The paper reports the effect of stress-induced martensite ageing (SIM-ageing), and subsequent training in the superelasticity temperature range (SE-training) upon compression, on the shape memory and superelastic response of [0 0 1]-oriented crystals of the Ti50Ni40Cu10 (at.%) alloy. It was shown that after SIM-ageing and SE-training, stabilized B19-martensite was observed. Stabilized B19-martensite does not suppress superelastic behavior, leads to development of martensitic transformation under loading with high hardening and creates the conditions for a compressive two-way shape memory effect of 0.9%. The magnitude of the normal shape memory effect was 3.2% at an external stress of 100 MPa and superelasticity up to 2–2.5%.