Understanding the explosion mechanism and hydrodynamic evolution of core-collapse supernovae (SNe) is a long-standing quest in astronomy. The asymmetries caused by the explosion are encoded into the line profiles which appear in the nebular phase of the SN evolution – with particularly clean imprints in He star explosions. Here, we carry out nine different supernova simulations of He-core progenitors, exploding them in 3D with parametrically varied neutrino luminosities using the PROMETHEUS-HOTB code, hydrodynamically evolving the models to the homologous phase. We then compute nebular phase spectra with the 3D Non-Local Thermodynamic Equilibrium spectral synthesis code EXTRASS (EXplosive TRAnsient Spectral Simulator). We study how line widths and shifts depend on progenitor mass, explosion energy, and viewing angle. We compare the predicted line profile properties against a large set of Type Ib observations, and discuss the degree to which current neutrino-driven explosions can match observationally inferred asymmetries. With self-consistent 3D modelling – circumventing the difficulties of representing 56Ni mixing and clumping accurately in 1D models – we find that neither low-mass He cores exploding with high energies nor high-mass cores exploding with low energies contribute to the Type Ib SN population. Models which have line profile widths in agreement with this population give sufficiently large centroid shifts for calcium emission lines. Calcium is more strongly affected by explosion asymmetries connected to the neutron star kicks than oxygen and magnesium. Lastly, we turn to the near-infrared spectra from our models to investigate the potential of using this regime to look for the presence of He in the nebular phase.