Special with regard what?
Are you sure about that? Is there no feature in the "governing laws" that might bring that into question?
There might be, but it is not necessarily the case for
all deterministic systems.
For
all deterministic systems, however, knowing the state and the laws at a given time one could establish any and every moment in the future.
But it is not true for all deterministic systems that knowing the state of the current or future state and the laws one can establish every moment in the past.
So yes, I am sure that knowing X+1 does not
necessarily mean you can know X, or any prior moment, in a deterministic system.
One only needs to have two different set of causes possibly leading to exactly the same output to understand that.
A simple deterministic system of squaring the previous number demonstrates the point adequately enough.
Are prior states special in the laws of physics?
In a deterministic system they, along with the laws, completely determine the current state, as explained, but it doesn't necessarily work in reverse.
So if you feel the need to deem them special, or as seems to be your want you want
me to deem them special, then they can be seen to be in that regard.
But I do not personally consider them particularly "special", and see no particular benefit of such a word in this context.
The states and moments are what they are.
In a deterministic system one leads inexorably and infallibly to the next.
If you wish to consider them special based on what I have offered, feel free, but you're going to have to clarify just exactly what you are asking if what I have so far said is insufficient for you.