There are several questions about the definition of Teichmuller space.
Definition(see A Primer on Mapping Class Groups)
Let $S$ be a genus g closed surface with $g \ge 2$.The Teichmuller space of $S$,$ Teich(S_g)$, is defined to be homotopic classes of marked hyperbolic surfaces. Two marked hyperbolic surfaces $(X_1,\varphi_1)$,$(X_2,\varphi_2)$ are said to be homotopic if and only if there is an isometry $I:X_1 \to X_2$ such that $I \circ \varphi_1$ is homotopic to $\varphi_2$ in the usual sense.
Let $S=T^2$ ,The Teichmuller space of $S$,$ Teich(T^2)$, is defined to be homotopic classes of marked unit-area flat surfaces on $T^2$.
Questions
In this definition, Are both the markings and the isometry required to be orientation$-$ preserving?
The followings are my understandings:
First of all, since we finally defined a Teichmuller metric on this space, the change of marking map must be orientation$-$ preserving, and thus the isometry in the definition must be orientation$-$ preserving. the requirement of orientation$-$ preserving for the isometry can also be seen from the fact that we always identify hyperbolic surfaces with Riemann surfaces and the isometry with biholomorphism.
Second, according to the definition for twist parameters $\theta_i$, markings should also be orientation$-$ preserving, otherwise, we can easily construct two marked hyperbolic surfaces with the same Fenchel-Nielsen Coordinates, which means the map $FN$ in the Theorem 10.6 is not injective.
All of those two points can actually be easily seen from the fact that the definition for twist parameters $\theta_i$ should be well$-$defined.
So we may suppose that those maps need to be orientation$-$ preserving. But the problem is that:
In another model of Teichmuller space, namely, $DF(\pi_1(S_g),PSL(2,R))/PGL(2,R)$, can we replace $PGL(2,R)$ by $PSL(2,R)$? If we insist all of maps constructed in the proof of Proposition 10.2 to be orientation$-$ preserving, then every conjugation will come from $PSL(2,R)$,rather than in $PGL(2,R)$ . More concretely, if there is a conjugation come from orientation$-$ reversing, just as the reflection appeared in the step 2 of the proof of Proposition 10.1, it is difficult for me to understand the corresponding orientation$-$ preserving marking and isometry in Teichmuller space.
Actually, this problem can also be explained in the definition of marked lattices,which required a choice of ordered generators ( ? $\blacktriangle$ But in the definition of high dimensional marked lattices in the page 354, we only need a choice of basis, and if we required an ordered basis, then $SL(2,R)$ acts $\clubsuit$ not transitively on the space of marked unit volume latices). $\bullet$ I don't know How to understand orientation$-$ reversing Euclidean isometries between marked lattices such as reflection about x$-$axis in the context of equivalence of marked hyperbolic surface under the assumptions that all maps are orientation$-$ preserving?( After all,in the second proof of Proposition 10.1, we only care orientation$-$ preserving linear map)
Assuming that all maps appeared in the definition of Teichmuller space are orientation$-$ preserving, then How to define the actions of $Mod^{\pm}(S)$ on $Teich(S_g)$ by $f \cdot (X,\varphi)=(X,\varphi \circ f^{-1})$, since we now don't have orientation$-$ reversing marking.
By Royden, $Mod(S)=Isom^{+}(Teich(S_g))$,but how to prove $Mod^{\pm}(S)=Isom(Teich(S_g))$?