We generated tunable 2-μm optical vortex pulses with a topological charge of 1 or 2 in the wavelength range 1.953-2.158 μm by realizing anisotropic transfer of the topological charge from the pump beam to the signal output in a vortex-pumped half-symmetric optical parametric oscillator. A maximum vortex output energy of 2.1 mJ was obtained at a pump energy of 22.8 mJ, which corresponds to a slope efficiency of 15%. The topological charges of the signal and idler output were investigated using a shearing interferometric technique employing a low-spatial-frequency transmission grating.