Time reversal symmetry in the Schrodinger equation and evolution of wave packet

My question has to do with time reversal symmetry and the arrow of time in quantum mechanics.

We know that the Schrodinger equation is invariant under time reversal. However, if we take an initial Gaussian wave function for a free particle, it always spreads forward in time. So the width of the Gaussian becomes bigger as time progresses, until the wave function is flat everywhere.

Doesn't this imply a preferred direction in time? For example, if a movie of spreading of probability density were played backward, one would see accumulation of probability around a small region of space instead of spreading. This obviously does not happen in nature. Also, wouldn't information be erased in such spreading? For example, if we take a completely flat wave packet (i.e. a small constant), we cannot evolve it back into a Gaussian using the Schrodinger equation.

Does this mean we can tell the direction of time by watching the evolution of a probability density? If so, how is this consistent with time reversal symmetry in QM?

No, this situation does not break time reversal symmetry, because the Gaussian also spreads out backwards in time. Its width is at a minimum at $t = 0$, and it increases as $t$ either increases or decreases. There's no preferred time direction at all.
One can intuitively understand this by looking at the Fourier components. The time $t = 0$ is the unique time all of the components "line up" to produce a purely real Gaussian.