Do Time-Asymmetric Laws call for Time-Asymmetric Spacetime Structure?

Many philosophers of physics take the failure of the laws of physics to be invariant under the time reversal transformation to give us good reason to think that spacetime is temporally anisotropic, yet the details of this inference are rarely made explicit. I discuss two reasonable ways of filling in the details of this inference, the first of which utilizes a symmetry principle proposed by John Earman and the second of which utilizes Harvey Brown’s account of spacetime. I contend that neither of the resulting arguments is sound.