When it comes to a Tduality invariant formulation of string theory, there are two primary actions that are useful to study as a point of entry. The first is Tseytlin’s noncovariant action. It is found in his formulation of the duality symmetric string, which presents a stringy extension of the FloreaniniJackiw Lagrangians for chiral fields. In fact, for the sigma model action in this formulation, one can directly reproduce the FloreaniniJackiw Lagrangians for antichiral and chiral scalar fields. The caveat is that, although we have explicit invariance, which is important because ultimately we want Tduality to be a manifest symmetry, we lose manifest Lorentz covariance on the string worldsheet. What one finds is that we must impose local Lorentz invariance onshell, and from this there are some interesting things to observe about the constraints imposed at the operator level.
The main papers to study are Tseytlin’s 1990/91 works listed below. Unfortunately there is no preprint available, so these now classic string papers remain buried behind a paywall:
1) Tseytlin, ‘Duality Symmetric Formulation of String World Sheet Dynamics‘
2) Tseytlin, ‘Duality Symmetric Closed String Theory and Interacting Chiral Scalars‘
For Hull’s doubled formalism, on the other hand, we have manifest 2dimensional invariance. In both cases the worldsheet action is formulated such that both the string coordinates and their duals are on equal footing, hence one thinks of the coordinates being doubled. However, one advantage in Hull’s formulation is that there is a priori doubling of the string coordinates in the target space. Here, invariance is effectively built in as a principle of construction. This is because for the covariant double sigma model action, the target space may be written as , in which we have a noncompact spacetime and a doubled torus. From the torus identifications we have manifest symmetry. Then after imposing what we define as the selfduality constraint of the theory, which contains an metric, invariance of the theory reduces directly to .

 Hull, ‘Doubled Geometry and TFolds‘
 Hull, ‘Geometry for Nongeometric Backgrounds‘
 Hull and ReidEdwards, ‘Nongeometric backgrounds, doubled geometry and generalised Tduality‘
What is neat about the two formulations is that, turning off interactions, they are found to be equivalent on a classical and quantum level. It is quite fun to work through them both and prove their equivalence, as it comes down to the constraints we must impose in both formulations.
I think the doubled formalism (following Hull) for sigma models is most interesting on a general level. I’m still not comfortable with different subtleties in the construction, for example the doubled torus fibration background or choice of polarisation from Tduality. The latter is especially curious. But, in the course of the last two weeks, things are finally beginning to clarify and I look forward to writing more about it in time.
Related to the above, I thought I’d share three other supplementary papers that I’ve found to be generally helpful:
1) Berman, Blair, Malek, and Perry, ‘O(D,D) Geometry of String Theory‘
2) Berman and Thompson, ‘Duality Symmetric String and Mtheory‘
3) Thompson, ‘Tduality Invariant Approaches to String Theory‘
There are of course many other papers, including stuff I’ve been studying on general double sigma models and relatedly the Pasti, Sorokin and Tonin method. But those listed above should be a good start for anyone with an itch of curiosity.