  Following recent advances in the local theory of current-algebraic orbifolds,
we study various geometric properties of the general WZW orbifold, the general
coset orbifold and a large class of (non-linear) sigma model orbifolds.
Phase-space geometry is emphasized for the WZW orbifolds - while for the sigma
model orbifolds we construct the corresponding {\it sigma model orbifold
action}, which includes the previously-known general WZW orbifold action and
general coset orbifold action as special cases. We focus throughout on the {\it
twisted Einstein tensors} with diagonal monodromy, including the twisted
Einstein metric, the twisted B field and the twisted torsion field of each
orbifold sector. Finally, we present strong evidence for a conjectured set of
{\it twisted Einstein equations} which should describe those sigma model
orbifolds in this class which are also 1-loop conformal.
