The Markov approximation is arguably the most ubiquitous tool in physics, underpinning quantum master equations, stochastic processes, and -- via Shannon's channel model and Lamport's logical clocks -- the foundational assumptions of distributed computing. It is widely assumed that Markovianity inherently implies temporal asymmetry: that the Markov property is a forward-in-time-only (FITO) construct. We show that this assumption is a category mistake in the sense of Ryle (1949). Guff, Shastry, and Rocco (2025) have recently demonstrated that the Markov approximation applied to the Caldeira-Leggett model -- a paradigmatic open quantum system -- maintains time-reversal symmetry in the derived equations of motion. The resulting time-symmetric formulations of quantum Brownian motion, Lindblad master equations, and Pauli master equations describe thermalisation that can occur in two opposing temporal directions. Asymmetry arises not from the dynamics but from boundary conditions. We trace how Markovianity's assumed directionality propagated from physics through Shannon's information theory to Lamport's happens-before relation and the impossibility theorems of distributed computing (FLP, CAP, Two Generals). Each step encodes FITO as convention, then treats it as physical law -- the same category mistake repeated across domains. The Surrey result establishes that this conflation is not merely philosophically suspect but mathematically unnecessary: the most fundamental approximation used to derive irreversibility is itself time-symmetric.