It is argued that the electron stripes as found in correlated oxides have t
o do with an unrecognized form of order. The manifestation of this order is
the robust property that the charge stripes are at the same time antiphase
boundaries in the spin system. We demonstrate that the quantity which is o
rdering is sublattice parity, referring to the geometric property of a bipa
rtite lattice that it can be subdivided in two sublattices in two different
ways. Reinterpreting standard results of one-dimensional physics, we demon
strate that the same order is responsible for the phenomenon of spin-charge
separation in strongly interacting one-dimensional electron systems. In fa
ct, the stripe phases can be seen from this perspective as the precise gene
ralization of the Luttinger liquid to higher dimensions. Most of this paper
is devoted to a detailed exposition of the mean-field theory of sublattice
parity order in 2 + 1 dimensions, Although the quantum dynamics of the spi
n and charge degrees of freedom are fully taken into account, a perfect sub
lattice parity order is imposed. Owing to novel order-out-of-disorder physi
cs, the sublattice parity order gives rise to full stripe order at long wav
elengths. This adds further credibility to the notion that stripes find the
ir origin in the microscopic quantum fluctuations and it suggests a novel v
iewpoint on the relationship between stripes and high-T-c superconductivity
.