  We discuss the construction of six- and four-dimensional Type IIB
orientifolds with vanishing untwisted RR tadpoles, but generically non-zero
twisted RR tadpoles. Tadpole cancellation requires the introduction of D-brane
systems with zero untwisted RR charge, but non-zero twisted RR charges. We
construct explicit models containing branes and antibranes at fixed points of
the internal space, or non-BPS branes partially wrapped on it. The models are
non-supersymmetric, but are absolutely stable against decay to supersymmetric
vacua. For particular values of the compactification radii tachyonic modes may
develop, triggering phase transitions between the different types of non-BPS
configurations of branes, which we study in detail in a particular example. As
an interesting spin-off, we show that the $\IT^6/\IZ_4$ orientifold without
vector structure, previously considered inconsistent due to uncancellable
twisted tadpoles, can actually be made consistent by introducing a set of
brane-antibrane pairs whose twisted charge cancels the problematic tadpole.
