An invariant based on ordered K-theory with coefficients in ℤ ⊕ ⊕_n1 ℤ/n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C*-algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C*-algebras in question. As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in ℤ⊕ ℚ⊕ (ℚ/ℤ. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.