  Stationary-state Schr{\"o}dinger-Pauli theory is a description of electrons
with a spin moment in an external electromagnetic field. For 2-electron systems
as described by the Schr{\"o}dinger-Pauli theory Hamiltonian with a symmetrical
binding potential, we report a new symmetry operation of the electronic
coordinates. The symmetry operation is such that it leads to the equality of
the transformed wave function to the wave function. This equality is referred
to as the Wave Function Identity. The symmetry operation is a two-step process:
an interchange of the spatial coordinates of the electrons whilst keeping their
spin moments unchanged, followed by an inversion. The Identity is valid for
arbitrary structure of the binding potential, arbitrary electron interaction of
the form $w(|{\bf{r}} - {\bf{r}}'|)$, all bound electronic states, and
arbitrary dimensionality. It is proved that the exact wave functions satisfy
the Identity. On application of the permutation operation for fermions to the
identity, it is shown that the parity of the singlet states is even and that of
triplet states odd. As a consequence, it follows that at electron-electron
coalescence, the singlet state wave functions satisfy the cusp coalescence
constraint, and triplet state wave functions the node coalescence condition.
Further, we show that the parity of the singlet state wave functions about all
points of electron-electron coalescence is even, and that of the triplet state
wave functions odd. The Wave Function Identity and the properties on parity,
together with the Pauli principle, are then elucidated by application to the
$2$-dimensional $2$-electron `artificial atoms' or semiconductor quantum dots
in a magnetic field in their first excited singlet $2^{1}S$ and triplet $2^{3}
S$ states.
