In this technical note, we introduce the study of the in-block controllability (IBC) of affine systems on polytopes. In particular, we study whether all the states in the interior of a given polytope are mutually accessible through its interior by applying uniformly bounded inputs. The proposed study formalizes controllability under state constraints, and it has strong connections to the study of controllability of piecewise affine hybrid systems. We provide easily checkable necessary conditions for IBC. Then, for an important class of polytopes, simplicial polytopes, we show that these conditions are also sufficient.