When watching a movie of a physical process, one can conjecture whether it is running forward or backward in time by examining key physical parameters in the process. Here, the authors show that superpositions of (thermodynamic) quantum processes with opposite time's arrows are also physically possible and observable, and explore the thermodynamic role played by the interference term. Microscopic physical laws are time-symmetric, hence, a priori there exists no preferential temporal direction. However, the second law of thermodynamics allows one to associate the "forward" temporal direction to a positive variation of the total entropy produced in a thermodynamic process, and a negative variation with its "time-reversal" counterpart. This definition of a temporal axis is normally considered to apply in both classical and quantum contexts. Yet, quantum physics admits also superpositions between forward and time-reversal processes, whereby the thermodynamic arrow of time becomes quantum-mechanically undefined. In this work, we demonstrate that a definite thermodynamic time's arrow can be restored by a quantum measurement of entropy production, which effectively projects such superpositions onto the forward (time-reversal) time-direction when large positive (negative) values are measured. Finally, for small values (of the order of plus or minus one), the amplitudes of forward and time-reversal processes can interfere, giving rise to entropy-production distributions featuring a more or less reversible process than either of the two components individually, or any classical mixture thereof.