  We describe square lattice spin liquids which break time-reversal symmetry,
while preserving translational symmetry. The states are distinguished by the
manner in which they transform under mirror symmetries. All the states have
non-zero scalar spin chirality, which implies the appearance of spontaneous
orbital charge currents in the bulk (even in the insulator); but in some cases,
orbital currents are non-zero only in a formulation with three orbitals per
unit cell. The states are formulated using both the bosonic and fermionic
spinon approaches. We describe states with $\mathbb{Z}_2$ and U(1) bulk
topological order, and the chiral spin liquid with semionic excitations. The
chiral spin liquid has no orbital currents in the one-band formulation, but
does have orbital currents in the three-band formulation. We discuss
application to the cuprate superconductors, after postulating that the broken
time-reversal and mirror symmetries persist into confining phases which may
also break other symmetries. In particular, the broken symmetries of the chiral
spin liquid could persist into the N\'eel state.
