Moduli space of torus compactifications I am trying to understand some general statements made in the lecture notes by Vafa entitled "Lectures on Strings and Dualities" concerning toroidal compactifications (arXiv:hep-th/9702201).
Question 1: First, consider the heterotic string (D = 10) bosonic string compactified on a torus $T^d$. It is stated on page 12 that the space of inequivalent choices of the left and right- moving momenta $(P_L, P_R)$ is given by the coset
$$\frac{SO(d,d)}{SO(d) \times SO(d) \times O(d,d;\mathbb{Z})}$$
I understand that this is basically $\frac{SO(d,d)}{SO(d) \times SO(d)}$ further modded out by the discrete T-duality group $O(d,d;\mathbb{Z})$ which corresponds to Lorentz boosts with integer coefficients. However, in the main text, it is stated that it is $O(d) \times O(d)$ transformations that do not change the string states. So strictly speaking, shouldn't we be modding $SO(d,d)$ by $O(d) \times O(d)$ instead of $SO(d) \times SO(d)$?
Question 2: Now consider $\mathcal{N} = 2$ theories on $T^d$. 
The following statement appears on page 13 about the type-IIA and type-IIB theories:

For more general compactification on $T^d$ the part of the T-duality group which does not exchange the two theories is $SO(d,d;\mathbb{Z})$; the elements of T-duality which are in $O(d,d;\mathbb{Z})$ but not in $SO(d,d;
\mathbb{Z})$ will exchange IIA and IIB and thus are not symmetries of either one.

So does this mean that the T-duality group consists of transformations that do take IIA to IIB (and vice-versa) but also transformations that do not take IIA to IIB and that the former correspond to $O(d,d;\mathbb{Z}) \setminus SO(d,d;\mathbb{Z})$? So when textbooks make statements like $IIA \leftrightarrow IIB$ under T-duality, they really mean under this set theoretic subtraction?
Question 3: This question, somewhat embarrasingly, is more about notation than physics. On page 13 of the same lecture notes, the following paragraph appears:

In compactifying type II strings on tori, the scalars parametrized by the coset $SO(d,d)/SO(d) \times SO(d)$ correspond to choices of the metric of the torus ($d(d+1)/2$ degrees of freedom) and the antisymmetric field $B_{ij}$ on the torus ($d(d-)1/2$ degrees of freedom).

Here, does the author mean $\frac{SO(d,d)}{SO(d) \times SO(d)}$ or does he mean $(SO(d,d)/SO(d)) \times SO(d)$, i.e. $\frac{SO(d,d)}{SO(d)} \times SO(d)$? The two are vastly different things.
In fact, often he refers to things like $SO(5,5)/SO(5) \times SO(5)$ or $SO(5,5)/SO(5) \times SO(5) \times SO(5,5;\mathbb{Z})$. What is the order in which these are to be read/interpreted?
 A: Regarding Question 1, you are talking about bosonic strings, rather than heterotic, since for heterotic strings left- and right- moving modes are compactified on $T^d$ and $T^{16+d}$ (or vice versa). Now, for bosonic strings on $T^d$ the moduli space should be $$M=\frac{O(d,d;\mathbb{R})}{O(d;\mathbb{R})\times O(d;\mathbb{R})\times O(d,d;\mathbb{Z})}$$
because the evenness condition is invariant under $O(d,d;\mathbb{R})$ transformations, and the mass formula $$m^2=2(p_L^2+p_R^2)+...$$ is invariant under $O(d;\mathbb{R})\times O(d;\mathbb{R})$ transformations (see Becker, Becker, Schwarz textbook, pages 270 and 283).
Question 2. As already answered by Danu, in type II superstrings, the transformations of $O(d,d;\mathbb{Z})$ with determinant $-1$ takes you between IIA and IIB (since they change relative chirality of fermions), whereas $det=+1$ transformations are symmetries within IIA and IIB separately (they don't change chiralities). So the moduli spaces of both type II theories (compactified on a torus) are the same.
As for Quation 3, the quotient by a product is the right one.
PS You probably already know the answers, but I'll leave it here anyway.
