  E. Segal proved that any autoequivalence of an enhanced triangulated category
can be realised as a spherical twist. However, when exhibiting an
autoequivalence as a spherical twist one has various choices for the source
category of the spherical functor. We describe a construction that realises the
composition of two spherical twists as the twist around a single spherical
functor whose source category semiorthogonally decomposes into the source
categories for the spherical functors we started with. We give a description of
the cotwist for this spherical functor and prove, in the special case when our
starting twists are around spherical objects, that the cotwist is the Serre
functor (up to a shift). We finish with an explicit treatment for the case of
P-objects.
