Does time-reversal turn the stable manifold of an equilibrium into unstable manifold?

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If a trajectory $x(t)$ starts in the stable manifold of an equilibrium, it tends to the equilibrium as $t\to +\infty$.

If we now consider the opposite direction of $t$ (i.e. draw the arrow that marks the direction in the opposite direction) then is the orbit in the unstable manifold of that equilibrium?