Twisted intertwining operators and tensor products of (generalized) twisted modules

We study the general twisted intertwining operators (intertwining operators among twisted modules) for a vertex operator algebra $V$. We give the skew-symmetry and contragredient isomorphisms between spaces of twisted intertwining operators and also prove some other properties of twisted intertwining operators. Using twisted intertwining operators, we introduce a notion of $P(z)$-tensor product of two objects for $z\in \mathbb{C}^{\times}$ in a category of suitable $g$-twisted $V$-modules for $g$ in a group of automorphisms of $V$ and give a construction of such a $P(z)$-tensor product under suitable assumptions. We also construct $G$-crossed commutativity isomorphisms and $G$-crossed braiding isomorphisms. We formulate a $P(z)$-compatibility condition and a $P(z)$-grading-restriction condition and use these conditions to give another construction of the $P(z)$-tensor product.