The 0(+) and 0(-) states of a model fermion system with symmetric and
antisymmetric spatial permutation symmetry have been investigated. The
pairwise interaction adopted has a repulsive core and an attractive t
ail. Evidences on the preferred shapes and on modes of internal motion
s have been extracted from the wave functions. A comparison is made am
ong the states having even or odd parity and being symmetric or antisy
mmetric. It is found that the symmetry plays an essential role in dete
rmining the microscopic structures.