  Genuinely two-dimensional robust crosscurrents -- which flow against the
natural direction of heat flux -- have been missing since the discovery of
their one-dimensional counterpart. We provide a setup to realize them on a
cubic three-dimensional (3D) lattice hosting a Hofstadter model coupled to two
heat baths with different temperatures. We show that these currents exhibit
dissipative robustness: they are stable against the presence of impurities and
tilting of the gauge field in certain nonequilibrium configurations. Moreover,
we find protected boundary currents with genuinely 3D robustness, i.e. they are
only stable if tunnelling can occur in all three spatial directions. The model
also presents generic surface currents, which are robust for both bosonic and
fermionic systems. We identify the underlying qualitative mechanism responsible
for the robustness of the surface currents and the crucial role played by
certain discrete symmetries.
