My understanding is that the large cardinals are indeed linearly ordered, which is a remarkable fact. Of course, it is not a theorem (nor could it ever be?), but merely an empirical fact; still, there is no known explanation for this completely surprising coincidence.

On the other hand, there are sentences $\sigma$ such that $T+\sigma>T$ and $T+\neg\sigma>T$. These are called Orey sentences and though they are rare, a number of them exist. Thus, $\le_{cons}$ is not linear, though the counterexamples are quite contrived (none of the known ones would be called "natural" by a mathematician).

Orey sentences are sometimes also called "Double jump sentences" and you can read about them (and find references) here:

http://plato.stanford.edu/entries/independence-large-cardinals/#IntHie

I hope this answers your question to your satisfaction!

My understanding is that the large cardinals are indeed linearly ordered, which is a remarkable fact. Of course, it is not a theorem (nor could it ever be?), but merely an empirical fact; still, there is no known explanation for this completely surprising coincidence.

On the other hand, there are sentences $\sigma$ such that $T+\sigma>T$ and $T+\neg\sigma>T$. These are called "Double jump sentences" and though they are rare, a number of them exist. Thus, $\le_{cons}$ is not linear, though the counterexamples are quite contrived (none of the known ones would be called "natural" by a mathematician).

You can read more about Double jump sentences (and find references) here:

http://plato.stanford.edu/entries/independence-large-cardinals/#IntHie

I hope this answers your question to your satisfaction!