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Non-Hermitian systems, with symmetric or antisymmetric Hamiltonians under the parity-time ($\mathcal{PT}$) operations, can have entirely real eigenvalues. This fact has led to surprising discoveries such as loss-induced lasing and topological energy transfer. A merit of anti-$\mathcal{PT}$ systems is free of gain, but in recent efforts on making anti-$\mathcal{PT}$ devices, nonlinearity is still required. Here, counterintuitively, we show how to achieve anti-$\mathcal{PT}$ symmetry and its spontaneous breaking in a linear device by spinning a lossy resonator. Compared with a Hermitian spinning device, significantly enhanced optical isolation and ultrasensitive nanoparticle sensing are achievable in the anti-$\mathcal{PT}$-broken phase. In a broader view, our work provides a new tool to study anti-$\mathcal{PT}$ physics, with such a wide range of applications as anti-$\mathcal{PT}$ lasers, anti-$\mathcal{PT}$ gyroscopes, and anti-$\mathcal{PT}$ topological photonics or optomechanics.