  Aimed to a deeper comprehension of a manifestly T-dual invariant formulation
of string theory, in this paper a detailed comparison between the non-covariant
action proposed by Tseytlin and the covariant one proposed by Hull is done.
These are obtained by making both the string coordinates and their duals
explicitly appear, on the same foot, in the world-sheet action, so "doubling"
the string coordinates along the compact dimensions. After a discussion of the
nature of the constraints in both the models and the relative quantization, it
results that the string coordinates and their duals behave like "non-commuting"
phase space type coordinates but their expressions in terms of Fourier modes
generate the oscillator algebra of the standard bosonic string formulation. A
proof of the equivalence of the two formulations is given. Furthermore,
open-string solutions are also discussed.
