By double torus I mean the connected sum of two torus $\mathbb{T}$ denoted $\mathbb{T} \# \mathbb{T}$. By triple projective space I mean the triple connected sum of the projective plane $\mathbb{RP}_2$, denoted $\mathbb{RP}_2 \# \mathbb{RP}_2 \# \mathbb{RP}_2$. The group action that I use is given by the identity and the antipode homeomorphism is $A:\mathbb{T} \# \mathbb{T} \to \mathbb{T} \# \mathbb{T}$ given by $p \mapsto -p$.
I'm told to take as a fundamental region a disk in the "border" that connects both surfaces and study the action of the antipode on that disk. My intuition gives however only one $\mathbb{RP}_2$. What's going on?
An image of the alleged disk:
Note that what I want to compute is $\mathbb{T} \# \mathbb{T}/ \{Id,A\}$.
Thoughts
Perhaps I could study first this antipode map for the simple torus $\mathbb{T}$ which gives the Klein bottle $\mathbb{RP}_2 \# \mathbb{RP}_2$. However, what result would allow me to generalize it for the double torus?