There are some results of Woodin that indicate why large cardinal axioms of a certain kind might indeed be linearly ordered by consistency strength. In Kanamori's chart there are some possibilities for nonlinearity, but I think this is just lack of knowledge.

I am not aware of any result that actually says that two statements are incomparable in consistency strength over ZF. Note however that this might be a problem of our current proof technology.

We have good technology to show that the consistency of a statement $\sigma$ implies the consistency of $\tau$ (all over ZF). Namely, start with a model of ZF+$\sigma$ and use forcing and/or inner models to construct a model of ZF+$\tau$. The second incompleteness theorem tells us how we can get a stronger theory from a consistent theory $T$, namely by adding the axiom "$T$ is consistent".

But how could we prove that two statements are incomparable in consistency strength?

There are some results of Woodin that indicate why large cardinal axioms of a certain kind might indeed be linearly ordered by consistency strength. In Kanamori's chart there are some possibilities for nonlinearity, but I think this is just lack of knowledge.

I am not aware of any result that actually says that two natural statements are incomparable in consistency strength over ZF. Note however that this might be a problem of our current proof technology.

We have good technology to show that the consistency of a statement $\sigma$ implies the consistency of $\tau$ (all over ZF). Namely, start with a model of ZF+$\sigma$ and use forcing and/or inner models to construct a model of ZF+$\tau$. The second incompleteness theorem tells us how we can get a stronger theory from a consistent theory $T$, namely by adding the axiom "$T$ is consistent".

But how could we prove that two given statements are incomparable in consistency strength?