The symmetry is the largest symmetry possible and hence the lowest entropy of any state.

We are enabled to perceive so clearly mostly by the low entropy gradient.

The separated layers represent a system of greater order and lower (not higher) entropy.

But perhaps regions of low entropy were merely a hint to some other kind of information.

In that state it has low entropy and contains little information.

So why is the observed entropy so low?

This information would have very low entropy.

Thus the second law follows, given initial conditions with low entropy.

But this does not address the question of why the initial state of the universe was that of low entropy.

The theory of cosmic inflation tries to give reason why the early universe had such a low entropy.