We precisely test scale invariance and examine local thermal equilibrium in the hydrodynamic expansion of a Fermi gas of atoms as a function of interaction strength. After release from an anisotropic optical trap, we observe that a resonantly interacting gas obeys scale-invariant hydrodynamics, where the mean square cloud size <r2>=<x2+y2+z2> expands ballistically (like a noninteracting gas) and the energy-averaged bulk viscosity is consistent with zero, 0.00(0.04)ℏn, with n the density. In contrast, the aspect ratios of the cloud exhibit anisotropic "elliptic" flow with an energy-dependent shear viscosity. Tuning away from resonance, we observe conformal symmetry breaking, where <r2> deviates from ballistic flow.