The usual Ricci flow and normalized Ricci flow for surfaces are $$ \partial_t g = -2Kg $$ and $$ \partial_t g = -2Kg + 2sg,$$ where $K$ is the Gaussian curvature and $s$ is its average. The latter equation can be solved on the interval $[0,\infty)$ and converges to a metric of constant Gaussian curvature.
Now consider the reverse flow $$\partial_t = 2Kg - 2sg,$$ which should function as some sort of reverse flow. What can you say about its long-time existence?
I had a discussion about this with someone, who argued that there should be no time-reversal of that flow as the flow would not know "where to go" if you start in, say, a sphere of constant curvature (because of the above convergence result; this flow should move the metric away from a constant curvature metric).
But clearly, at least for round spheres the above equation has a unique solution, as $\partial_t g = 0$. So why can't a round sphere, for instance, be a repelling fix point of the flow... Writing this, I realize that this flow, if it existed, would have one crucial disadvantage: $g(t)$ would not depend continuously on $g$ in the $C^k$-topology.