  CP asymmetries for neutrino oscillations in matter can be disentangled into
the matter-induced CPT-odd (T-invariant) component and the genuine T-odd
(CPT-invariant) component. For their understanding in terms of the relevant
ingredients, we develop a new perturbative expansion in both $\Delta
m^2_{21},\, |a| \ll |\Delta m^2_{31}|$ without any assumptions between $\Delta
m^2_{21}$ and $a$, and study the subtleties of the vacuum limit in the two
terms of the CP asymmetry, moving from the CPT-invariant vacuum limit $a \to 0$
to the T-invariant limit $\Delta m^2_{21} \to 0$. In the experimental region of
terrestrial accelerator neutrinos, we calculate their approximate expressions
from which we prove that, at medium baselines, the CPT-odd component is small
and nearly $\delta$-independent, so it can be subtracted from the experimental
CP asymmetry as a theoretical background, provided the hierarchy is known. At
long baselines, on the other hand, we find that (i) a Hierarchy-odd term in the
CPT-odd component dominates the CP asymmetry for energies above the first
oscillation node, and (ii) the CPT-odd term vanishes, independent of the CP
phase $\delta$, at $E =0.92~\mathrm{GeV}\,(L/1300~\mathrm{km})$ near the second
oscillation maximum, where the T-odd term is almost maximal and proportional to
$\sin\delta$. A measurement of the CP asymmetry in these energy regions would
thus provide separate information on (i) the neutrino mass ordering, and (ii)
direct evidence of genuine CP violation in the lepton sector.
