  We study four dimensional $Z_2 \times Z_2$ (shift)-orientifolds in presence
of internal magnetic fields and NS-NS $B$-field backgrounds, describing in some
detail one explicit example with N=1 supersymmetry. These models are related by
$T$-duality to orientifolds with $D$-branes intersecting at angles and exhibit,
due to the background fields, a rank reduction of the gauge group and multiple
matter families. Moreover, the low-energy spectra are chiral and anomaly free
if $D5$-branes are present along the magnetized directions.
