Suppose I have a smooth projective variety $X$ and a semiorthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mutations to the full asmissible triangulated subcats that make up the semiorth decomposition. Algorithmically it is clear to me what happens, but what is the spirit of such a transformation? what is the geometry behind it? in particular I would be curious to understand this in the case of full exceptional sequences and when all subcats in the semiorth decomposition are generated by sheaves, line or vector bundles.

I wouldn't say that in general there is some special geometry behind mutations. This just explains that whenever you have one sod, you have many of them. However, in some cases there is some meaning. For example, if $D(X) = < A, {}^\perp A >$ then the left mutation functor $L_A:{}^\perp A \to A^\perp$ is isomorphic to the composition of the Serre functor of $D(X)$ and the inverse Serre functor of ${}^\perp A$: $$ L_A = S_X\circ S_{{}^\perp A}^{1}. $$ In the particular case, when ${}^\perp A$ is generated by one exceptional object, one has $L_A = S_X$. 

