If $T = \langle A ,B\rangle$ is a semiorthogonal decomposition and $\alpha_\ast:A \to T$, $\beta_\ast:B \to T$ are the embedding functors then one can consider the functor $\beta^\ast\circ\alpha_\ast:A \to B$ (or its right adjoint $\alpha^!\circ\beta_\ast:B \to A$). This is called the gluing functor. Morally, the ways of gluing $A$ to $B$ are classified by gluing functors --- to each functor one should be able to associate a triangulated category $T$ with a s.o.d. into $A$ and $B$ for which the gluing functor is isomorphic to the given one. Because of the well-known problems with nonfunctoriality of the cone, one cannot hope to prove this precise statement. However, if everything has a DG-enhancement, one can.

Indeed, assume that $A = Hot(R)$, $B = Hot(S)$, where $R$ and $S$ are pretriangulated DG-algebras and assume that the functor $A \to B$ is realized by a $R-S$-bimodule $M$. Then one can consider a DG-algebra of the form
$$
U = \left(\begin{array}{cc} R & 0 \cr M & S \end{array}\right).
$$
Then $T := Hot(U)$ should give what you want.