  In this paper we completely describe the order isomorphisms between cones of
atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an
algebraic direct summand of the so-called engaged and disengaged part. On the
cone of the engaged part every order isomorphism is linear and the disengaged
part consists only of copies of $\mathbb{R}$. Furthermore, in the setting of
general JB-algebras we prove the following. If either algebra does not contain
an ideal of codimension one, then every order isomorphism between their cones
is linear if and only if it extends to a homeomorphism, between the cones of
the atomic part of their biduals, for a suitable weak topology.
