  We show that a quantum field theory A living on the line and having a group G
of inner symmetries gives rise to a category GLoc A of twisted representations.
This category is a braided crossed G-category in the sense of Turaev. Its
degree zero subcategory is braided and equivalent to the usual representation
category Rep A. Combining this with [29], where Rep A was proven to be modular
for a nice class of rational conformal models, and with the construction of
invariants of G-manifolds in [60], we obtain an equivariant version of the
following chain of constructions: Rational CFT -> modular category ->
3-manifold invariant.
  We then study the relation between GLoc A and the braided (in the usual
sense) representation category Rep A^G of the orbifold theory A^G. We prove the
equivalence Rep A^G = (GLoc A)^G, which is a rigorous implementation of the
insight that one needs to take the twisted representations of A into account in
order to determine Rep A^G. In the opposite direction we have GLoc A = Rep A^G
\rtimes S, where S \subset Rep A^G is the full subcategory of representations
of A^G contained in the vacuum representation of A, and \rtimes refers to the
Galois extensions of braided tensor categories of [44,48].
  If A is completely rational and G is finite we prove that A has g-twisted
representations for every g in G. In the holomorphic case (where Rep A =
Vect_C) this allows to classify the possible categories GLoc A and to clarify
the role of the twisted quantum doubles D^\omega(G) in this context, as will be
done in a sequel. We conclude with some remarks on non-holomorphic orbifolds
and surprising counterexamples concerning permutation orbifolds.
