  This paper is to study what we call twisted regular representations for
vertex operator algebras. Let $V$ be a vertex operator algebra, let
$\sigma_1,\sigma_2$ be commuting finite-order automorphisms of $V$ and let
$\sigma=(\sigma_1\sigma_2)^{-1}$. Among the main results, for any
$\sigma$-twisted $V$-module $W$ and any nonzero complex number $z$, we
construct a weak $\sigma_1\otimes \sigma_2$-twisted $V\otimes V$-module
$\mathfrak{D}_{\sigma_1,\sigma_2}^{(z)}(W)$ inside $W^{*}$. Let $W_1,W_2$ be
$\sigma_1$-twisted, $\sigma_2$-twisted $V$-modules, respectively. We show that
$P(z)$-intertwining maps from $W_1\otimes W_2$ to $W^{*}$ are the same as
homomorphisms of weak $\sigma_1\otimes \sigma_2$-twisted $V\otimes V$-modules
from $W_1\otimes W_2$ into $\mathfrak{D}_{\sigma_1,\sigma_2}^{(z)}(W)$. We also
show that a $P(z)$-intertwining map from $W_1\otimes W_2$ to $W^{*}$ is
equivalent to an intertwining operator of type $\binom{W'}{W_1\; W_2}$, which
is a twisted version of a result of Huang and Lepowsky. Finally, we show that
for each $\tau$-twisted $V$-module $M$ with $\tau$ any finite-order
automorphism of $V$, the coefficients of the $q$-graded trace function lie in
$\mathfrak{D}_{\tau,\tau^{-1}}^{(-1)}(V)$, which generate a $\tau\otimes
\tau^{-1}$-twisted $V\otimes V$-submodule isomorphic to $M\otimes M'$.
