Let's say we have two 4-manifolds with boundary $M_1$ and $M_2$, with spin-$\mathbb{C}$ structures $s_1$ and $s_2$. Let's say also there is an orientation reversing diffeomorphism $f:\partial M_1 \to \partial M_2$. I heard in a talk that there is an induced spin-$\mathbb{C}$ structure on $M_1\cup_{f}M_2$, denoted $s_1\cup s_2$. How can one construct such structure? Is there a hands-on way to see how this works?
By the Mayer-Vietoris sequence and Poincaré duality, we get a map $H^2(M_1,\partial M_1, \mathbb{Z})\oplus H^2(M_2, \partial M_2,\mathbb{Z})\to H^2(M_1\cup_{f}M_2,\mathbb{Z})$. I understand that spin-$\mathbb{C}$ structures are in correspondence with second cohomology because we generate any structure by twisting a fixed one by a complex line bundle. However, this does not quite give me an understanding of how this spin-$\mathbb{C}$ structure $s_1\cup s_2$ is being constructed from $s_1$ and $s_2$. Hence my question.
There are many different ways to define spin-$\mathbb{C}$ structures, but the one that comes to me first is a hermitian 4-dimensional bundle $S=S^+\oplus S^-$ on $M$ with a Clifford multiplication $\rho:TM\to \mathfrak{su}(S)\subset Hom(S,S)$ (i.e., each vector field acts as a traceless, skew-adjoint endomorphism on $S$; we can extend this to make sense of Clifford multiplication by forms).