Leonard Susskind, in lecture 2 of his course on classical mechanics, provides two different descriptions of reversibility. The first description (given at video time 3:00) is that if a system is in a given state then, using the laws that govern the time-evolution of the system, you know with certainty the state it was in at some earlier time. He later says that this can be thought of in another way (at video time 49:00): the laws that govern time-evolution (in this context just $\mathbf{F}=m\mathbf{a}$) are invariant under the transformation $t\mapsto -t$.
Are these descriptions saying the same thing, or is the 2nd description just one way of satisfying the first description? Stated another way, the 2nd description is clearly sufficient for the 1st description, but is it necessary? To flesh this out a bit, $\mathbf{F}=m\mathbf{a}$ is deterministic. That is, if you know the state of a system at a given time and the forces that act on it, you know with certainty the state it will be in at some later time. So, if you make the $t\mapsto -t$ transformation, then it is (loosely speaking) deterministic in the reverse time sense; i.e., you know the state of the system in the past. That clearly satisfies the first notion of reversibility. But, it seems plausible that there might be other ways to know the past with certainty without the laws of physics being time-reversal invariant. To clarify, by plausible I'm not asserting that I think there are other ways to satisfy the 1st notion of reversibility, simply that I can't convince myself this is the only way to satisfy it.
These two descriptions seem to me to be saying qualitatively different things (at least at a superficial level). The first one seems to say that given the present you know the past, whereas the second seems to say that it's impossible to distinguish the forward and reverse time directions based solely on $\mathbf{F}=m\mathbf{a}$.
