  We construct new explicit vacuum solutions of quadratic metric-affine
gravity. The approach of metric-affine gravity in using an independent affine
connection produces a theory with 10+64 unknowns, which implies admitting
torsion and possible nonmetricity. Our spacetimes are generalisations of
classical pp-waves, four-dimensional Lorentzian spacetimes which admit a
nonvanishing parallel spinor field. We generalize this definition to metric
compatible spacetimes with pp-metric and purely axial torsion. It has been
suggested that one can interpret that the axial component of torsion as the
Hodge dual of the electromagnetic vector potential. We compare these solutions
with our previous results and other solutions of classical models describing
the interaction of gravitational and neutrino fields.
