A rotary electric machine, e.g., a cycloidal reluctance motor, includes a stator having stator teeth connected to a cylindrical stator core, and a rotor having a cylindrical rotor core. The stator core and/or rotor core are constructed of grain-oriented, spirally-wound ferrous material having a circular or annular grain orientation. The stator teeth may be constructed of grain-oriented steel having a linear grain orientation. Notches may be spaced around an inner circumferential surface of the stator core, with each stator tooth engaged with a respective notch. The rotor may be eccentrically positioned radially within the stator. The rotor core may define notches spaced around its outer circumferential surface, with salient rotor projections engaged with a respective rotor notch. The machine in such an embodiment may be a switched reluctance rotor. An electrical system using the machine and a method of manufacturing the machine are also disclosed.