general relativity - In GR, how do particles know how to fall in instead of out of a gravitational well?

The geodesic equation (let's suppose that we're talking about massive particles, so I'll parameterize the path by proper time $\tau$)

$\frac{d^2 x^\mu}{d \tau^2} + \Gamma^\mu_{\rho \sigma}\frac{d x^\rho}{d \tau} \frac{d x^\sigma}{d \tau}=0$

is invariant under $\tau \rightarrow -\tau$.

However, falling particles clearly have a direction, they always fall in instead of out. Formally, how does a falling particle 'know' which way to move in $\tau$?