We report a double-sided two-mirror neodymium-doped KGd(WO4)2/Nd:KGW conical refraction laser which produces a linearly polarised output from the flat output coupler and a cone-refracted output from the curved output coupler. The linearly polarised output had a polarisation extinction ratio of 20:1 with a Gaussian beam profile and a measured M
2=1. The unpolarised output had the intensity and polarisation distribution corresponding to a cone-refracted Gaussian beam with extinction ratio of 2.7:1.