  We study relativistic hydrodynamics of normal fluids in two spatial
dimensions. When the microscopic theory breaks parity, extra transport
coefficients appear in the hydrodynamic regime, including the Hall viscosity,
and the anomalous Hall conductivity. In this work we classify all the transport
coefficients in first order hydrodynamics. We then use properties of response
functions and the positivity of entropy production to restrict the possible
coefficients in the constitutive relations. All the parity-breaking transport
coefficients are dissipationless, and some of them are related to the
thermodynamic response to an external magnetic field and to vorticity. In
addition, we give a holographic example of a strongly interacting relativistic
fluid where the parity-violating transport coefficients are computable.
