  We study some aspects of asymmetric orbifolds of tori, with the orbifold
group being some $\mathbb{Z}_N$ subgroup of the T-duality group and, in
particular, provide a concrete understanding of certain phase factors that may
accompany the T-duality operation on the stringy Hilbert space in toroidal
compactification. We discuss how these T-duality twist phase factors are
related to the symmetry and locality properties of the closed string vertex
operator algebra, and clarify the role that they enact in the modular
covariance of the orbifold theory, mainly using asymmetric orbifolds of tori
which are root lattices as working examples.
