  In order to support the odd moduli in models of (type IIB) string
compactification, we classify the Calabi-Yau threefolds with h^{1,1}<=4 which
exhibit pairs of identical divisors, with different line-bundle charges,
mapping to each other under possible divisor exchange involutions. For this
purpose, the divisors of interest are identified as completely rigid surface,
Wilson surface, K3 surface and some other deformation surfaces. Subsequently,
various possible exchange involutions are examined under the symmetry of
Stanley-Reisner Ideal. In addition, we search for the Calabi-Yau theefolds
which contain a divisor with several disjoint components. Under certain
reflection involution, such spaces also have nontrivial odd components in
(1,1)-cohomology class. String compactifications on such Calabi-Yau
orientifolds with non-zero h^{1,1}_-(CY_3/\sigma) could be promising for
concrete model building in both particle physics and cosmology. In the spirit
of using such Calabi-Yau orientifolds in the context of LARGE volume scenario,
we also present some concrete examples of (strong/weak) swiss-cheese type
volume form.
