A new X-ray undulator has been designed and constructed which produces linearly polarized X-rays in which the plane of polarization can be oriented to a user selectable angle, from horizontal to vertical. Based on the Apple-II elliptically polarizing undulator (EPU), the undulator rotates the angle of the linear polarization by a simple longitudinal motion of the undulator magnets. Combined with the circular and elliptical polarization capabilities of the EPU operating in the standard mode, this new undulator produces soft X-ray radiation with versatile polarization control. This paper describes the magnetic structure of the device and presents an analysis of the magnetic ®eld with varying undulator parameters. The variable linear polarization capability is then exhibited by measuring the X-ray absorption spectrum of an oriented polytetra¯uoroethylene thin ®lm. This experiment, which measures the linear dichroism of the sample at two peaks near the C 1s absorption edge, demonstrates the continuous polarization rotation capabilities of the undulator.