The grooving technique was employed for manufacturing a self-standing curved Ge crystal. The crystal focuses hard X-rays with high efficiency by diffraction in Laue geometry through asymmetric bent planes. The sample was tested at the Institut Laue-Langevin (Grenoble, France), undergoing two types of characterization. A monochromatic and low-divergence gamma-ray beam was used to test the curvature of asymmetric planes, showing a diffraction performance better than for any mosaic crystal under equal conditions. Then, the focusing capability of the crystal was probed through a polychromatic and fine-focus hard X-ray beam. Asymmetric (220) planes were chosen for analysis because of the impossibility to obtain a curvature along this family of planes via any symmetric configuration in focusing crystals. A method for calculating the curvatures induced in any family of lattice planes is also presented.