(I don't know how to ask this other than by laying out my worldview. Needless to say, this is here to be dissected and disemboweled. I realize the broad sweeps will irk people. I hope that by presenting this question as a weak hypothesis, fuzzy conceptions about equivalence, information and order will be challenged, in turn answering the question in the title.)
This has bugged me for more than 20 years. To me, equivalence is the fundamental currency of logic, information and all order.
Some branches of logic don't deal in absolutes, but they have equivalence as their limit or minima (as in AI local minima).
A single bit is an equivalence determination. Bit-data establishes property equivalence, when it already exists.
I've read in one statistics or philosophy book or-other that equivalence and equivalence relations underpin all of set theory, which I understand underpins most if not all mathematics.
In my view, equality is the most fundamental "unit" in nature. The most basic causally relevant state.
If equality is a causally relevant state, "order" is a higher-level (based on usage) state of equivalence. Whether order initiated by natural selection, in spontaneous chemical processes or even driven by gravity, it is the state of property equivalence. The Moon is an orderly arrangement of matter held together by gravity. It is matter with equivalent properties (position and momentum).
Subatomic, atomic and molecular structures represent property equivalence that can be disrupted by heat.
So property equivalence describes the building blocks of matter as with particles, atoms and molecules. Those property equivalences lead to higher levels of order and ultimately to information.
I believe information is too broadly defined. In this treatment, it is closer to the end of the chain.
As an example of overuse of the word information: Yes quantum information exists, but only if we perform calculations with it. It is an overstatement to say that all spontaneous quantum processes are informational.
This last point is to weaken information's grip it has on virtually everthing. Being loosely and nearly universally applied, it disrupts our understanding of more fundamental processes. It is much more revealing to consider how organismal information flow has equivalence determinations at its foundation.
For example, actual neural networks are probabilistic. It is easier to visualize a hierarchy of equivalence (math-like) steps rather than "information" flowing through brain circuits.