  We analyze the brane content and charges in all of the orientifold string
theories on space-times of the form E x R^8, where E is an elliptic curve with
holomorphic or anti-holomorphic involution. Many of these theories involve
"twistings" coming from the B-field and/or sign choices on the orientifold
planes. A description of these theories from the point of view of algebraic
geometry, using the Legendre normal form, naturally divides them into three
groupings. The physical theories within each grouping are related to one
another via sequences of T-dualities. Our approach agrees with both previous
topological calculations of twisted KR-theory and known physics arguments, and
explains how the twistings originate from both a mathematical and a physical
perspective.
